A097852 -id:A097852

A097851

G.f.: (1+x^8+x^9+x^10+x^18)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

+10
14

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 20, 22, 31, 36, 47, 54, 71, 80, 102, 117, 144, 164, 201, 227, 272, 309, 365, 411, 483, 540, 627, 702, 806, 898, 1026, 1137, 1289, 1427, 1606, 1770, 1985, 2179, 2429, 2663, 2952, 3225, 3565, 3882, 4272, 4644, 5090, 5518, 6032, 6522

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Poincare series for invariant polynomial functions on the space of binary forms of degree 8.

LINKS

Table of n, a(n) for n=0..55.

Andries Brouwer, Poincaré Series (See n=8).

J.-I. Igusa, Modular forms and projective invariants, Amer. J. Math., 89 (1967), 817-855; see p. 847.

Peter Littelmann and Claudio Procesi, On the Poincaré series of the invariants of binary forms, Journal of Algebra 133.2 (1990): 490-499. See last page.

Index entries for linear recurrences with constant coefficients, signature (-1,1,3,3,0,-3,-4,-3,-1,1,2,3,3,2,1,-1,-3,-4,-3,0,3,3,1,-1,-1).

PROG

(PARI) Vec((1+x^8+x^9+x^10+x^18)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)) + O(x^50)) \\ Jinyuan Wang, Mar 10 2020

CROSSREFS

For these Poincare series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

This Poincare series is mentioned in A079293.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Sep 01 2004

STATUS

approved

A293933

Poincaré series for invariant polynomial functions on the space of binary forms of degree 7.

+10
13

1, 0, 1, 0, 4, 0, 10, 4, 18, 13, 35, 26, 62, 52, 97, 92, 153, 144, 229, 223, 325, 329, 456, 460, 624, 636, 826, 856, 1084, 1119, 1398, 1449, 1766, 1845, 2214, 2306, 2743, 2860, 3349, 3507, 4065, 4245, 4889, 5107, 5820, 6093, 6893, 7200, 8108

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Many of these Poincaré series have every other term zero, in which case these zeros have been omitted.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Andries Brouwer, Poincaré Series (See n=7)

FORMULA

a(n) = (11/8640)*n^4 + (11/1080)*n^3 + O(n^2). - Robert Israel, Oct 20 2017

EXAMPLE

The Poincaré series is (1 - t^6 + 2t^8 - t^10 + 5t^12 + 2t^14 + 6t^16 + 2t^18 + 5t^20 - t^22 + 2t^24 - t^26 + t^32) / (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)

MAPLE

(x^16-x^13+2*x^12-x^11+5*x^10+2*x^9+6*x^8+2*x^7+5*x^6-x^5+2*x^4-x^3+1)/(-x^2+1)/(-x^3+1)/(-x^4+1)/(-x^5+1)/(-x^6+1);

f := gfun:-rectoproc({-12*a(n) - 60*a(n+1) - 168*a(n+2) - 348*a(n+3) - 588*a(n+4) - 852*a(n+5) - 1080*a(n+6) - 1212*a(n+7) - 1212*a(n+8) - 1080*a(n+9) - 852*a(n+10) - 588*a(n+11) - 348*a(n+12) - 168*a(n+13) - 60*a(n+14) - 12*a(n+15) + 11*n^4 + 418*n^3 + 6433*n^2 + 46778*n + 136380, a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 0, a(4) = 4, a(5) = 0, a(6) = 10, a(7) = 4, a(8) = 18, a(9) = 13, a(10) = 35, a(11) = 26, a(12) = 62, a(13) = 52, a(14) = 97, a(15) = 92, a(16) = 153}, a(n), remember):

map(f, [$0..100]); # Robert Israel, Oct 20 2017

MATHEMATICA

a = DifferenceRoot[Function[{a, n},

{-60*a[n + 1] - 168*a[n + 2] -

348*a[n + 3] - 588*a[n + 4] -

852*a[n + 5] - 1080*a[n + 6] -

1212*a[n + 7] - 1212*a[n + 8] -

1080*a[n + 9] - 852*a[n + 10] -

588*a[n + 11] - 348*a[n + 12] -

168*a[n + 13] - 60*a[n + 14] -

12*a[n + 15] - 12*a[n] + 11*n^4 +

418*n^3 + 6433*n^2 + 46778*n + 136380 == 0,

a[0] == 1, a[1] == 0, a[2] == 1,

a[3] == 0, a[4] == 4, a[5] == 0,

a[6] == 10, a[7] == 4, a[8] == 18,

a[9] == 13, a[10] == 35, a[11] == 26,

a[12] == 62, a[13] == 52, a[14] == 97}]];

Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2019, after Robert Israel *)

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293934

Poincaré series for invariant polynomial functions on the space of binary forms of degree 9.

+10
13

1, 0, 2, 0, 8, 5, 28, 27, 84, 99, 217, 273, 506, 647, 1066, 1367, 2082, 2649, 3811, 4796, 6612, 8228, 10960, 13483, 17487, 21274, 26979, 32490, 40443, 48242, 59107, 69885, 84470, 99074, 118330, 137762, 162842, 188287, 220516, 253377, 294316, 336213, 387658, 440463, 504484

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..44.

Andries Brouwer, Poincaré Series (See n=9)

EXAMPLE

The Poincaré series is (1 + t^4 - t^6 + 5t^8 + 3t^10 + 18t^12 + 15t^14 + 44t^16 + 43t^18 + 82t^20 + 76t^22 + 122t^24 + 107t^26 + 147t^28 + 119t^30 + 147t^32 + 107t^34 + 122t^36 + 76t^38 + 82t^40 + 43t^42 + 44t^44 + 15t^46 + 18t^48 + 3t^50 + 5t^52 - t^54 + t^56 + t^60) / (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)(1 - t^14)(1 - t^16)

MAPLE

nmax := 120 :

(1 + t^4 - t^6 + 5*t^8 + 3*t^10 + 18*t^12 + 15*t^14 + 44*t^16 + 43*t^18 + 82*t^20 + 76*t^22 + 122*t^24 + 107*t^26 + 147*t^28 + 119*t^30 + 147*t^32 + 107*t^34 + 122*t^36 + 76*t^38 + 82*t^40 + 43*t^42 + 44*t^44 + 15*t^46 + 18*t^48 + 3*t^50 + 5*t^52 - t^54 + t^56 + t^60) / (1 - t^4)/(1 - t^6)/(1 - t^8)/(1 - t^10)/(1 - t^12)/(1 - t^14)/(1 - t^16) ;

taylor(%, t=0, nmax) :

gfun[seriestolist](%) :

seq( %[1+2*i], i=0..nmax/2-1) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293935

Poincaré series for invariant polynomial functions on the space of binary forms of degree 10.

+10
13

1, 0, 1, 0, 2, 0, 6, 0, 12, 5, 24, 13, 52, 33, 97, 80, 177, 160, 319, 301, 540, 547, 887, 926, 1429, 1512, 2219, 2402, 3367, 3681, 5015, 5502, 7294, 8064, 10419, 11550, 14664, 16253, 20287, 22531, 27682, 30738, 37319, 41378, 49671, 55060, 65390, 72391, 85250

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..48.

Andries Brouwer, Poincaré Series (See n=10)

EXAMPLE

The Poincaré series is (1 - t^5 + 2t^6 - t^7 + 4t^8 + 4t^9 + 8t^10 + 6t^11 + 16t^12 + 9t^13 + 17t^14 + 15t^15 + 19t^16 + 12t^17 + 23t^18 + 12t^19 + 19t^20 + 15t^21 + 17t^22 + 9t^23 + 16t^24 + 6t^25 + 8t^26 + 4t^27 + 4t^28 - t^29 + 2t^30 - t^31 + t^36) / (1 - t^2)(1 - t^4)(1 - t^5)(1 - t^6)^2(1 - t^7)(1 - t^8)(1 - t^9)

MAPLE

nmax := 120 :

(1 - t^5 + 2*t^6 - t^7 + 4*t^8 + 4*t^9 + 8*t^10 + 6*t^11 + 16*t^12 + 9*t^13 + 17*t^14 + 15*t^15 + 19*t^16 + 12*t^17 + 23*t^18 + 12*t^19 + 19*t^20 + 15*t^21 + 17*t^22 + 9*t^23 + 16*t^24 + 6*t^25 + 8*t^26 + 4*t^27 + 4*t^28 - t^29 + 2*t^30 - t^31 + t^36) / (1 - t^2)/(1 - t^4)/(1 - t^5)/(1 - t^6)^2/(1 - t^7)/(1 - t^8)/(1 - t^9) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ;

seq( %[1+i], i=0..nmax/2-1) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293936

Poincaré series for invariant polynomial functions on the space of binary forms of degree 11.

+10
13

1, 0, 2, 0, 13, 13, 73, 110, 320, 529, 1160, 1893, 3569, 5633, 9600, 14636, 23246, 34233, 51618, 73716, 106678, 148325, 207652, 281963, 384141, 510882, 680155, 888201, 1159414, 1489689, 1911698, 2421067, 3060774, 3826640, 4774174, 5899837, 7274612, 8895981, 10853618, 13146996

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..39.

Andries Brouwer, Poincaré Series (See n=11)

EXAMPLE

The Poincaré series is (1 + t^4 - t^6 + 10t^8 + 11t^10 + 58t^12 + 85t^14 + 222t^16 + 336t^18 + 660t^20 + 951t^22 + 1589t^24 + 2154t^26 + 3188t^28 + 4080t^30 + 5510t^32 + 6633t^34 + 8310t^36 + 9443t^38 + 11059t^40 + 11894t^42 + 13094t^44 + 13319t^46 + 13852t^48 + 13319t^50 + 13094t^52 + 11894t^54 + 11059t^56 + 9443t^58 + 8310t^60 + 6633t^62 + 5510t^64 + 4080t^66 + 3188t^68 + 2154t^70 + 1589t^72 + 951t^74 + 660t^76 + 336t^78 + 222t^80 + 85t^82 + 58t^84 + 11t^86 + 10t^88 - t^90 + t^92 + t^96) / (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)(1 - t^14)(1 - t^16)(1 - t^18)(1 - t^20)

MAPLE

nmax := 120 :

(1 + t^4 - t^6 + 10*t^8 + 11*t^10 + 58*t^12 + 85*t^14 + 222*t^16 + 336*t^18 + 660*t^20 + 951*t^22 + 1589*t^24 + 2154*t^26 + 3188*t^28 + 4080*t^30 + 5510*t^32 + 6633*t^34 + 8310*t^36 + 9443*t^38 + 11059*t^40 + 11894*t^42 + 13094*t^44 + 13319*t^46 + 13852*t^48 + 13319*t^50 + 13094*t^52 + 11894*t^54 + 11059*t^56 + 9443*t^58 + 8310*t^60 + 6633*t^62 + 5510*t^64 + 4080*t^66 + 3188*t^68 + 2154*t^70 + 1589*t^72 + 951*t^74 + 660*t^76 + 336*t^78 + 222*t^80 + 85*t^82 + 58*t^84 + 11*t^86 + 10*t^88 - t^90 + t^92 + t^96) / (1 - t^4)/(1 - t^6)/(1 - t^8)/(1 - t^10)/(1 - t^12)/(1 - t^14)/(1 - t^16)/(1 - t^18)/(1 - t^20) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ;

seq( %[1+2*i], i=0..nmax/2-1) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293937

Poincaré series for invariant polynomial functions on the space of binary forms of degree 12.

+10
13

1, 0, 1, 1, 3, 3, 8, 10, 20, 28, 52, 73, 127, 181, 291, 418, 639, 902, 1330, 1848, 2634, 3603, 4998, 6718, 9113, 12058, 16027, 20903, 27307, 35123, 45198, 57412, 72874, 91519, 114762, 142605, 176883, 217679, 267324, 326073, 396837, 480074, 579460, 695704, 833361, 993548

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..45.

Andries Brouwer, Poincaré Series (See n=12)

EXAMPLE

The Poincaré series is (1 + t^4 + t^5 + 3t^6 + 4t^7 + 7t^8 + 9t^9 + 17t^10 + 21t^11 + 36t^12 + 45t^13 + 65t^14 + 81t^15 + 110t^16 + 131t^17 + 168t^18 + 193t^19 + 232t^20 + 256t^21 + 293t^22 + 307t^23 + 336t^24 + 339t^25 + 351t^26 + 339t^27 + 336t^28 + 307t^29 + 293t^30 + 256t^31 + 232t^32 + 193t^33 + 168t^34 + 131t^35 + 110t^36 + 81t^37 + 65t^38 + 45t^39 + 36t^40 + 21t^41 + 17t^42 + 9t^43 + 7t^44 + 4t^45 + 3t^46 + t^47 + t^48 + t^52) / (1 - t^2)(1 - t^3)(1 - t^4)(1 - t^5)(1 - t^6)(1 - t^7)(1 - t^8)(1 - t^9)(1 - t^10)(1 - t^11)

MAPLE

nmax := 120 :

(1 + t^4 + t^5 + 3*t^6 + 4*t^7 + 7*t^8 + 9*t^9 + 17*t^10 + 21*t^11 + 36*t^12 + 45*t^13 + 65*t^14 + 81*t^15 + 110*t^16 + 131*t^17 + 168*t^18 + 193*t^19 + 232*t^20 + 256*t^21 + 293*t^22 + 307*t^23 + 336*t^24 + 339*t^25 + 351*t^26 + 339*t^27 + 336*t^28 + 307*t^29 + 293*t^30 + 256*t^31 + 232*t^32 + 193*t^33 + 168*t^34 + 131*t^35 + 110*t^36 + 81*t^37 + 65*t^38 + 45*t^39 + 36*t^40 + 21*t^41 + 17*t^42 + 9*t^43 + 7*t^44 + 4*t^45 + 3*t^46 + t^47 + t^48 + t^52) / (1 - t^2)/(1 - t^3)/(1 - t^4)/(1 - t^5)/(1 - t^6)/(1 - t^7)/(1 - t^8)/(1 - t^9)/(1 - t^10)/(1 - t^11) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943, respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293938

Poincaré series for invariant polynomial functions on the space of binary forms of degree 13.

+10
13

1, 0, 2, 0, 22, 33, 181, 375, 1120, 2342, 5467, 10668, 21660, 39562, 72816, 125484, 215161, 352424, 572086, 897867, 1394315, 2110350, 3159826, 4635480, 6731131, 9612072, 13595657, 18964299, 26221103, 35828058, 48562922, 65155439, 86777107, 114549589, 150198041, 195400674, 252651242

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..36.

Andries Brouwer, Poincaré Series (See n=13)

EXAMPLE

The Poincaré series is (1 + t^4 - t^6 + 19t^8 + 31t^10 + 157t^12 + 321t^14 + 885t^16 + 1756t^18 + 3794t^20 + 6856t^22 + 12788t^24 + 21324t^26 + 35633t^28 + 55326t^30 + 85174t^32 + 124064t^34 + 178645t^36 + 246238t^38 + 334814t^40 + 439321t^42 + 568305t^44 + 712862t^46 + 881834t^48 + 1061455t^50 + 1259989t^52 + 1459221t^54 + 1666984t^56 + 1860904t^58 + 2049854t^60 + 2209072t^62 + 2349306t^64 + 2446352t^66 + 2514111t^68 + 2530530t^70 + 2514111t^72 + 2446352t^74 + 2349306t^76 + 2209072t^78 + 2049854t^80 + 1860904t^82 + 1666984t^84 + 1459221t^86 + 1259989t^88 + 1061455t^90 + 881834t^92 + 712862t^94 + 568305t^96 + 439321t^98 + 334814t^100 + 246238t^102 + 178645t^104 + 124064t^106 + 85174t^108 + 55326t^110 + 35633t^112 + 21324t^114 + 12788t^116 + 6856t^118 + 3794t^120 + 1756t^122 + 885t^124 + 321t^126 + 157t^128 + 31t^130 + 19t^132 - t^134 + t^136 + t^140)/ (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)(1 - t^14)(1 - t^16)(1 - t^18)(1 - t^20)(1 - t^22)(1 - t^24)

MAPLE

nmax := 120 :

(1 + t^4 - t^6 + 19*t^8 + 31*t^10 + 157*t^12 + 321*t^14 + 885*t^16 + 1756*t^18 + 3794*t^20 + 6856*t^22 + 12788*t^24 + 21324*t^26 + 35633*t^28 + 55326*t^30 + 85174*t^32 + 124064*t^34 + 178645*t^36 + 246238*t^38 + 334814*t^40 + 439321*t^42 + 568305*t^44 + 712862*t^46 + 881834*t^48 + 1061455*t^50 + 1259989*t^52 + 1459221*t^54 + 1666984*t^56 + 1860904*t^58 + 2049854*t^60 + 2209072*t^62 + 2349306*t^64 + 2446352*t^66 + 2514111*t^68 + 2530530*t^70 + 2514111*t^72 + 2446352*t^74 + 2349306*t^76 + 2209072*t^78 + 2049854*t^80 + 1860904*t^82 + 1666984*t^84 + 1459221*t^86 + 1259989*t^88 + 1061455*t^90 + 881834*t^92 + 712862*t^94 + 568305*t^96 + 439321*t^98 + 334814*t^100 + 246238*t^102 + 178645*t^104 + 124064*t^106 + 85174*t^108 + 55326*t^110 + 35633*t^112 + 21324*t^114 + 12788*t^116 + 6856*t^118 + 3794*t^120 + 1756*t^122 + 885*t^124 + 321*t^126 + 157*t^128 + 31*t^130 + 19*t^132 - t^134 + t^136 + t^140)/ (1 - t^4)/(1 - t^6)/(1 - t^8)/(1 - t^10)/(1 - t^12)/(1 - t^14)/(1 - t^16)/(1 - t^18)/(1 - t^20)/(1 - t^22)/(1 - t^24) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ;

seq( %[1+2*i], i=0..nmax/2-1) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293939

Poincaré series for invariant polynomial functions on the space of binary forms of degree 14.

+10
13

1, 0, 1, 0, 3, 0, 10, 4, 31, 27, 97, 110, 291, 375, 802, 1111, 2077, 2930, 5034, 7120, 11463, 16133, 24737, 34435, 50866, 69904, 100123, 135828, 189639, 253889, 346959, 458657, 615228, 803647, 1060615, 1369871, 1782235, 2277690, 2925659, 3702394, 4701294, 5894945, 7408011

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..42.

Andries Brouwer, Poincaré Series (See n=14)

EXAMPLE

The Poincaré series is (1 + t^4 - t^5 + 5t^6 + 3t^7 + 18t^8 + 21t^9 + 56t^10 + 72t^11 + 155t^12 + 209t^13 + 375t^14 + 523t^15 + 836t^16 + 1131t^17 + 1695t^18 + 2234t^19 + 3132t^20 + 4029t^21 + 5371t^22 + 6691t^23 + 8566t^24 + 10348t^25 + 12736t^26 + 14971t^27 + 17789t^28 + 20306t^29 + 23400t^30 + 25973t^31 + 29023t^32 + 31385t^33 + 34068t^34 + 35858t^35 + 37893t^36 + 38831t^37 + 39932t^38 + 39890t^39 + 39932t^40 + 38831t^41 + 37893t^42 + 35858t^43 + 34068t^44 + 31385t^45 + 29023t^46 + 25973t^47 + 23400t^48 + 20306t^49 + 17789t^50 + 14971t^51 + 12736t^52 + 10348t^53 + 8566t^54 + 6691t^55 + 5371t^56 + 4029t^57 + 3132t^58 + 2234t^59 + 1695t^60 + 1131t^61 + 836t^62 + 523t^63 + 375t^64 + 209t^65 + 155t^66 + 72t^67 + 56t^68 + 21t^69 + 18t^70 + 3t^71 + 5t^72 - t^73 + t^74 + t^78) / (1 - t^2)(1 - t^4)(1 - t^5)(1 - t^6)^2(1 - t^7)(1 - t^8)(1 - t^9)(1 - t^10)(1 - t^11)(1 - t^12)(1 - t^13)

MAPLE

nmax := 120 :

(1 + t^4 - t^5 + 5*t^6 + 3*t^7 + 18*t^8 + 21*t^9 + 56*t^10 + 72*t^11 + 155*t^12 + 209*t^13 + 375*t^14 + 523*t^15 + 836*t^16 + 1131*t^17 + 1695*t^18 + 2234*t^19 + 3132*t^20 + 4029*t^21 + 5371*t^22 + 6691*t^23 + 8566*t^24 + 10348*t^25 + 12736*t^26 + 14971*t^27 + 17789*t^28 + 20306*t^29 + 23400*t^30 + 25973*t^31 + 29023*t^32 + 31385*t^33 + 34068*t^34 + 35858*t^35 + 37893*t^36 + 38831*t^37 + 39932*t^38 + 39890*t^39 + 39932*t^40 + 38831*t^41 + 37893*t^42 + 35858*t^43 + 34068*t^44 + 31385*t^45 + 29023*t^46 + 25973*t^47 + 23400*t^48 + 20306*t^49 + 17789*t^50 + 14971*t^51 + 12736*t^52 + 10348*t^53 + 8566*t^54 + 6691*t^55 + 5371*t^56 + 4029*t^57 + 3132*t^58 + 2234*t^59 + 1695*t^60 + 1131*t^61 + 836*t^62 + 523*t^63 + 375*t^64 + 209*t^65 + 155*t^66 + 72*t^67 + 56*t^68 + 21*t^69 + 18*t^70 + 3*t^71 + 5*t^72 - t^73 + t^74 + t^78) / (1 - t^2)/(1 - t^4)/(1 - t^5)/(1 - t^6)^2/(1 - t^7)/(1 - t^8)/(1 - t^9)/(1 - t^10)/(1 - t^11)/(1 - t^12)/(1 - t^13) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293940

Poincaré series for invariant polynomial functions on the space of binary forms of degree 15.

+10
13

1, 0, 3, 1, 36, 80, 418, 1111, 3581, 8899, 22786, 51286, 114049, 234754, 472443, 902625, 1683916, 3024451, 5313062, 9063638, 15158162, 24760532, 39743317, 62563090, 96977396, 147874275, 222433862, 329908935, 483445738, 699822112, 1002221943, 1419949064, 1992553143

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..32.

Andries Brouwer, Poincaré Series (See n=915)

EXAMPLE

The Poincaré series is (1 + 2t^4 + 32t^8 + 76t^10 + 378t^12 + 995t^14 + 3048t^16 + 7294t^18 + 17681t^20 + 37736t^22 + 78903t^24 + 152321t^26 + 285968t^28 + 507762t^30 + 876759t^32 + 1451423t^34 + 2341739t^36 + 3653241t^38 + 5568497t^40 + 8254649t^42 + 11983447t^44 + 16987847t^46 + 23631274t^48 + 32196429t^50 + 43116834t^52 + 56681420t^54 + 73342055t^56 + 93320393t^58 + 117007543t^60 + 144461993t^62 + 175919353t^64 + 211175615t^66 + 250222591t^68 + 292516508t^70 + 337751801t^72 + 385016863t^74 + 433713649t^76 + 482605505t^78 + 530877973t^80 + 577086324t^82 + 620343376t^84 + 659172312t^86 + 692798202t^88 + 719914717t^90 + 740045690t^92 + 752239053t^94 + 756462172t^96 + 752239053t^98 + 740045690t^100 + 719914717t^102 + 692798202t^104 + 659172312t^106 + 620343376t^108 + 577086324t^110 + 530877973t^112 + 482605505t^114 + 433713649t^116 + 385016863t^118 + 337751801t^120 + 292516508t^122 + 250222591t^124 + 211175615t^126 + 175919353t^128 + 144461993t^130 + 117007543t^132 + 93320393t^134 + 73342055t^136 + 56681420t^138 + 43116834t^140 + 32196429t^142 + 23631274t^144 + 16987847t^146 + 11983447t^148 + 8254649t^150 + 5568497t^152 + 3653241t^154 + 2341739t^156 + 1451423t^158 + 876759t^160 + 507762t^162 + 285968t^164 + 152321t^166 + 78903t^168 + 37736t^170 + 17681t^172 + 7294t^174 + 3048t^176 + 995t^178 + 378t^180 + 76t^182 + 32t^184 + 2t^188 + t^192) / (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)(1 - t^14)(1 - t^16) (1 - t^18)(1 - t^20)(1 - t^22)(1 - t^24)(1 - t^26)(1 - t^28)

MAPLE

nmax := 120 :

(1 + 2*t^4 + 32*t^8 + 76*t^10 + 378*t^12 + 995*t^14 + 3048*t^16 + 7294*t^18 + 17681*t^20 + 37736*t^22 + 78903*t^24 + 152321*t^26 + 285968*t^28 + 507762*t^30 + 876759*t^32 + 1451423*t^34 + 2341739*t^36 + 3653241*t^38 + 5568497*t^40 + 8254649*t^42 + 11983447*t^44 + 16987847*t^46 + 23631274*t^48 + 32196429*t^50 + 43116834*t^52 + 56681420*t^54 + 73342055*t^56 + 93320393*t^58 + 117007543*t^60 + 144461993*t^62 + 175919353*t^64 + 211175615*t^66 + 250222591*t^68 + 292516508*t^70 + 337751801*t^72 + 385016863*t^74 + 433713649*t^76 + 482605505*t^78 + 530877973*t^80 + 577086324*t^82 + 620343376*t^84 + 659172312*t^86 + 692798202*t^88 + 719914717*t^90 + 740045690*t^92 + 752239053*t^94 + 756462172*t^96 + 752239053*t^98 + 740045690*t^100 + 719914717*t^102 + 692798202*t^104 + 659172312*t^106 + 620343376*t^108 + 577086324*t^110 + 530877973*t^112 + 482605505*t^114 + 433713649*t^116 + 385016863*t^118 + 337751801*t^120 + 292516508*t^122 + 250222591*t^124 + 211175615*t^126 + 175919353*t^128 + 144461993*t^130 + 117007543*t^132 + 93320393*t^134 + 73342055*t^136 + 56681420*t^138 + 43116834*t^140 + 32196429*t^142 + 23631274*t^144 + 16987847*t^146 + 11983447*t^148 + 8254649*t^150 + 5568497*t^152 + 3653241*t^154 + 2341739*t^156 + 1451423*t^158 + 876759*t^160 + 507762*t^162 + 285968*t^164 + 152321*t^166 + 78903*t^168 + 37736*t^170 + 17681*t^172 + 7294*t^174 + 3048*t^176 + 995*t^178 + 378*t^180 + 76*t^182 + 32*t^184 + 2*t^188 + t^192) / (1 - t^4)/(1 - t^6)/(1 - t^8)/(1 - t^10)/(1 - t^12)/(1 - t^14)/(1 - t^16) /(1 - t^18)/(1 - t^20)/(1 - t^22)/(1 - t^24)/(1 - t^26)/(1 - t^28) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ;

seq( %[1+2*i], i=0..nmax/2-1) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

A293941

Poincaré series for invariant polynomial functions on the space of binary forms of degree 16.

+10
13

1, 0, 1, 1, 3, 4, 13, 18, 47, 84, 177, 320, 639, 1120, 2077, 3581, 6235, 10395, 17344, 27940, 44848, 70180, 108921, 165817, 250256, 371558, 546960, 794363, 1143783, 1628190, 2299144, 3214042, 4459495, 6133164, 8375820, 11349269, 15278595, 20423345, 27136816, 35827488, 47037493, 61397294, 79726515, 102977471, 132370606, 169322488, 215620140, 273339320, 345063648, 433787088, 543198659, 677563207, 842079818, 1042751237, 1286826668, 1582652314, 1940231900, 2371051392, 2888771603, 3509044867, 4250358055, 5133832789, 6184270777, 7429930460

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.

LINKS

Table of n, a(n) for n=0..63.

Andries Brouwer, Poincaré Series (See n=16)

EXAMPLE

The Poincaré series is (1 + t^4 + 2t^5 + 8t^6 + 11t^7 + 28t^8 + 51t^9 + 102t^10 + 177t^11 + 340t^12 + 561t^13 + 980t^14 + 1586t^15 + 2565t^16 + 3955t^17 + 6095t^18 + 8991t^19 + 13206t^20 + 18815t^21 + 26498t^22 + 36437t^23 + 49596t^24 + 66028t^25 + 87003t^26 + 112578t^27 + 144034t^28 + 181363t^29 + 226014t^30 + 277437t^31 + 337179t^32 + 404317t^33 + 479951t^34 + 562691t^35 + 653453t^36 + 749807t^37 + 852481t^38 + 958443t^39 + 1067723t^40 + 1176799t^41 + 1285637t^42 + 1389850t^43 + 1489589t^44 + 1580460t^45 + 1662409t^46 + 1731403t^47 + 1788102t^48 + 1828489t^49 + 1854175t^50 + 1862064t^51 + 1854175t^52 + 1828489t^53 + 1788102t^54 + 1731403t^55 + 1662409t^56 + 1580460t^57 + 1489589t^58 + 1389850t^59 + 1285637t^60 + 1176799t^61 + 1067723t^62 + 958443t^63 + 852481t^64 + 749807t^65 + 653453t^66 + 562691t^67 + 479951t^68 + 404317t^69 + 337179t^70 + 277437t^71 + 226014t^72 + 181363t^73 + 144034t^74 + 112578t^75 + 87003t^76 + 66028t^77 + 49596t^78 + 36437t^79 + 26498t^80 + 18815t^81 + 13206t^82 + 8991t^83 + 6095t^84 + 3955t^85 + 2565t^86 + 1586t^87 + 980t^88 + 561t^89 + 340t^90 + 177t^91 + 102t^92 + 51t^93 + 28t^94 + 11t^95 + 8t^96 + 2t^97 + t^98 + t^102) / (1 - t^2)(1 - t^3)(1 - t^4)(1 - t^5)(1 - t^6)(1 - t^7) (1 - t^8)(1 - t^9)(1 - t^10)(1 - t^11)(1 - t^12)(1 - t^13)(1 - t^14) (1 - t^15)

MAPLE

nmax := 120 :

(1 + t^4 + 2*t^5 + 8*t^6 + 11*t^7 + 28*t^8 + 51*t^9 + 102*t^10 + 177*t^11 + 340*t^12 + 561*t^13 + 980*t^14 + 1586*t^15 + 2565*t^16 + 3955*t^17 + 6095*t^18 + 8991*t^19 + 13206*t^20 + 18815*t^21 + 26498*t^22 + 36437*t^23 + 49596*t^24 + 66028*t^25 + 87003*t^26 + 112578*t^27 + 144034*t^28 + 181363*t^29 + 226014*t^30 + 277437*t^31 + 337179*t^32 + 404317*t^33 + 479951*t^34 + 562691*t^35 + 653453*t^36 + 749807*t^37 + 852481*t^38 + 958443*t^39 + 1067723*t^40 + 1176799*t^41 + 1285637*t^42 + 1389850*t^43 + 1489589*t^44 + 1580460*t^45 + 1662409*t^46 + 1731403*t^47 + 1788102*t^48 + 1828489*t^49 + 1854175*t^50 + 1862064*t^51 + 1854175*t^52 + 1828489*t^53 + 1788102*t^54 + 1731403*t^55 + 1662409*t^56 + 1580460*t^57 + 1489589*t^58 + 1389850*t^59 + 1285637*t^60 + 1176799*t^61 + 1067723*t^62 + 958443*t^63 + 852481*t^64 + 749807*t^65 + 653453*t^66 + 562691*t^67 + 479951*t^68 + 404317*t^69 + 337179*t^70 + 277437*t^71 + 226014*t^72 + 181363*t^73 + 144034*t^74 + 112578*t^75 + 87003*t^76 + 66028*t^77 + 49596*t^78 + 36437*t^79 + 26498*t^80 + 18815*t^81 + 13206*t^82 + 8991*t^83 + 6095*t^84 + 3955*t^85 + 2565*t^86 + 1586*t^87 + 980*t^88 + 561*t^89 + 340*t^90 + 177*t^91 + 102*t^92 + 51*t^93 + 28*t^94 + 11*t^95 + 8*t^96 + 2*t^97 + t^98 + t^102) / (1 - t^2)/(1 - t^3)/(1 - t^4)/(1 - t^5)/(1 - t^6)/(1 - t^7) /(1 - t^8)/(1 - t^9)/(1 - t^10)/(1 - t^11)/(1 - t^12)/(1 - t^13)/(1 - t^14) /(1 - t^15) ;

taylor(%, t=0, nmax) ;

gfun[seriestolist](%) ; # R. J. Mathar, Oct 26 2017

CROSSREFS

For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved