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Search: a219837 -id:a219837
Displaying 1-9 of 9 results found.
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A219751
Expansion of x^4*(2-3*x-x^2)/((1+x)*(1-2*x)^2).
+10
10
0, 0, 0, 0, 2, 3, 8, 16, 36, 76, 164, 348, 740, 1564, 3300, 6940, 14564, 30492, 63716, 132892, 276708, 575260, 1194212, 2475804, 5126372, 10602268, 21903588, 45205276, 93206756, 192005916, 395196644, 812762908, 1670265060, 3430008604, 7038974180, 14435862300
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO], (2012), p. 14 (Lemma 4.3).
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).
FORMULA
G.f.: x^4*(2-3*x-x^2)/((1+x)*(1-2*x)^2).
a(n) = (2^(n-5)*(3*n+16)+4*(-1)^n)/9 with n>3, a(0)=a(1)=a(2)=a(3)=0. [Bruno Berselli, Nov 29 2012]
MATHEMATICA
CoefficientList[Series[x^4 (2 - 3 x - x^2)/((1 + x) (1 - 2 x)^2), {x, 0, 35}], x] (* Bruno Berselli, Nov 30 2012 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(2-3*x-x^2)/((1+x)*(1-2*x)^2), x, 0, n), x, n), n, 0, 35); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 2, 3, 8]; [n le 7 select I[n] else 3*Self(n-1) - 4*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Dec 14 2012
CROSSREFS
Cf. A219752-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219759
Expansion of x^4*(2-12*x+24*x^2-8*x^3-41*x^4+57*x^5-16*x^6)/((1-x)*(1-3*x+x^2)*(1-2*x)^6).
+10
10
0, 0, 0, 0, 2, 20, 120, 570, 2355, 8841, 30906, 102187, 323053, 984354, 2908671, 8375521, 23594410, 65237027, 177520325, 476515378, 1264297431, 3321423193, 8653113914, 22386784603, 57586262493, 147447786562, 376173191919, 957113924753, 2430649701066
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO] (2012), p. 30.
Index entries for linear recurrences with constant coefficients, signature (16,-112,449,-1132,1852,-1952,1264,-448,64).
FORMULA
G.f.: x^4*(2-12*x+24*x^2-8*x^3-41*x^4+57*x^5-16*x^6)/((1-x)*(1-3*x+x^2)*(1-2*x)^6).
MAPLE
A219759 := proc(n)
if n <= 1 then
0;
else
2^n*(881*n^2/24-14393*n/60+137+7*n^4/24-49*n^3/8+n^5/120) -384+ 64*A001906(n+2) ;
%/64 ;
end if;
end proc:
seq(A219759(n), n=0..20) ; # R. J. Mathar, Aug 19 2022
MATHEMATICA
CoefficientList[Series[x^4 (2 - 12 x + 24 x^2 - 8 x^3 - 41 x^4 + 57 x^5 - 16 x^6)/((1 - x) (1 - 3 x + x^2) (1 - 2 x)^6), {x, 0, 28}], x] (* Bruno Berselli, Nov 30 2012 *)
LinearRecurrence[{16, -112, 449, -1132, 1852, -1952, 1264, -448, 64}, {0, 0, 0, 0, 2, 20, 120, 570, 2355, 8841, 30906}, 40] (* Harvey P. Dale, Mar 01 2023 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(2-12*x+24*x^2-8*x^3-41*x^4+57*x^5-16*x^6)/((1-x)*(1-3*x+x^2)*(1-2*x)^6), x, 0, n), x, n), n, 0, 28); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 2, 20, 120, 570, 2355, 8841, 30906, 102187, 323053]; [n le 11 select I[n] else 16*Self(n-1) -112*Self(n-2) + 449*Self(n-3) - 1132*Self(n-4) + 1852*Self(n-5) - 1952*Self(n-6) + 1264*Self(n-7) - 448*Self(n-8) + 64*Self(n-9): n in [1..30]]; // Vincenzo Librandi, Dec 14 2012
CROSSREFS
Cf. A219751-A219758, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219754
Expansion of x^4*(1-x-x^2)/((1+x)*(1-2*x)*(1-x-2*x^2)).
+10
4
0, 0, 0, 0, 1, 1, 4, 7, 18, 37, 84, 179, 390, 833, 1784, 3791, 8042, 16989, 35804, 75243, 157774, 330105, 689344, 1436935, 2990386, 6213781, 12893604, 26719267, 55302678, 114333617, 236123784, 487160639, 1004147450, 2067947213, 4255199084, 8749007451, 17975233502
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO] (2012), p. 17 (Lemma 4.5).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).
FORMULA
G.f.: x^4*(1-x-x^2)/((1+x)^2*(1-2*x)^2).
a(n) = (2^(n-4)*(3*n+5)+(3*n-2)*(-1)^n)/27 for n=1 and n>2, a(0)=a(1)=a(2)=0. [Bruno Berselli, Nov 29 2012]
MATHEMATICA
CoefficientList[Series[x^4 (1 - x - x^2)/((1 + x) (1 - 2 x) (1 - x - 2 x^2)), {x, 0, 36}], x] (* Bruno Berselli, Nov 30 2012 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(1-x-x^2)/((1+x)*(1-2*x)*(1-x-2*x^2)), x, 0, n), x, n), n, 0, 36); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 1, 1, 4, 7]; [n le 8 select I[n] else 2*Self(n-1) + 3*Self(n-2) - 4*Self(n-3) - 4*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219752
Expansion of 2*x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)).
+10
3
0, 0, 0, 0, 2, 4, 12, 26, 62, 136, 302, 654, 1412, 3018, 6422, 13592, 28662, 60230, 126212, 263810, 550222, 1145352, 2380062, 4938078, 10230852, 21169114, 43749862, 90317816, 186263462, 383769046, 790000452, 1624890194, 3339501662, 6858353128, 14075255822
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO], (2012), p. 17 (Lemma 4.4).
Index entries for linear recurrences with constant coefficients, signature (4,-2,-7,4,4).
FORMULA
G.f.: 2*x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)).
a(n) = 2*A219753(n). - Bruno Berselli, Nov 29 2012
MATHEMATICA
CoefficientList[Series[2 x^4 (1 - 2 x + x^4)/((1 + x) (1 - 2 x)^2 (1 - x - x^2)), {x, 0, 34}], x] (* Bruno Berselli, Nov 30 2012 *)
LinearRecurrence[{4, -2, -7, 4, 4}, {0, 0, 0, 0, 2, 4, 12, 26, 62}, 40] (* Harvey P. Dale, Mar 09 2023 *)
PROG
(Maxima) makelist(coeff(taylor(2*x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)), x, 0, n), x, n), n, 0, 34); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 2, 4, 12, 26, 62]; [n le 9 select I[n] else 4*Self(n-1) - 2*Self(n-2) - 7*Self(n-3) + 4*Self(n-4) + 4*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 14 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219753
Expansion of x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)).
+10
2
0, 0, 0, 0, 1, 2, 6, 13, 31, 68, 151, 327, 706, 1509, 3211, 6796, 14331, 30115, 63106, 131905, 275111, 572676, 1190031, 2469039, 5115426, 10584557, 21874931, 45158908, 93131731, 191884523, 395000226, 812445097, 1669750831, 3429176564, 7037627911, 14433683991
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-7,4,4).
FORMULA
G.f.: x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)) (half of A219752).
a(n) = (2^(n-5)*(3*n+16)+4*(-1)^n)/9+((1-sqrt(5))^(n-3)-(1+sqrt(5))^(n-3))/(2^(n-3)*sqrt(5)) with n>3, a(0)=a(1)=a(2)=a(3)=0. [Bruno Berselli, Nov 29 2012]
a(n) = A219751(n)-A000045(n-3), n>=4. - R. J. Mathar, Aug 19 2022
MAPLE
A219753 := proc(n)
if n < 4 then
0 ;
else
128*(-1)^n+2^n*(16+3*n)-288*A000045(n-3) ;
%/288 ;
end if ;
end proc:
seq(A219753(n), n=0..20) ; # R. J. Mathar, Aug 19 2022
MATHEMATICA
CoefficientList[Series[x^4 (1 - 2 x + x^4)/((1 + x) (1 - 2 x)^2 (1 - x - x^2)), {x, 0, 35}], x] (* Bruno Berselli, Nov 30 2012 *)
LinearRecurrence[{4, -2, -7, 4, 4}, {0, 0, 0, 0, 1, 2, 6, 13, 31}, 40] (* Harvey P. Dale, Oct 05 2021 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)), x, 0, n), x, n), n, 0, 35); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 1, 2, 6, 13, 31]; [n le 9 select I[n] else 4*Self(n-1) - 2*Self(n-2) - 7*Self(n-3) + 4*Self(n-4) + 4*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 14 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219758
Expansion of x^4*(2-6*x+5*x^2+2*x^3-4*x^4)/((1-x)^2*(1-2*x)^5).
+10
2
0, 0, 0, 0, 2, 18, 99, 432, 1641, 5674, 18315, 56076, 164621, 466958, 1287183, 3463184, 9125905, 23617554, 60162067, 151126036, 374931477, 919863318, 2234253335, 5377622040, 12836667417, 30410801178, 71546437659, 167252066332, 388677763101, 898319253534
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO] (2012), p. 26.
Index entries for linear recurrences with constant coefficients, signature (12,-61,170,-280,272,-144,32).
FORMULA
G.f.: x^4*(2-6*x+5*x^2+2*x^3-4*x^4)/((1-x)^2*(1-2*x)^5).
a(n) = 2^(n-7)*(n^4-2*n^3-25*n^2+194*n-576)/3 +n+1 with n>1, a(0)=a(1)=0. [Bruno Berselli, Nov 29 2012]
MATHEMATICA
CoefficientList[Series[x^4 (2 - 6 x + 5 x^2 + 2 x^3 - 4 x^4)/((1 - x)^2 (1 - 2 x)^5), {x, 0, 29}], x] (* Bruno Berselli, Nov 30 2012 *)
LinearRecurrence[{12, -61, 170, -280, 272, -144, 32}, {0, 0, 0, 0, 2, 18, 99, 432, 1641}, 30] (* Harvey P. Dale, Jul 07 2017 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(2-6*x+5*x^2+2*x^3-4*x^4)/((1-x)^2*(1-2*x)^5), x, 0, n), x, n), n, 0, 29); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 2, 18, 99, 432, 1641, 5674]; [n le 10 select I[n] else 12*Self(n-1) - 61*Self(n-2) + 170*Self(n-3) - 280*Self(n-4) + 272*Self(n-5) - 144*Self(n-6) + 32*Self(n-7): n in [1..30]]; // Vincenzo Librandi, Dec 14 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219755
Expansion of x^4*(1-3*x^2-x^3)/((1+x)*(1-2*x)*(1-x-2*x^2)).
+10
1
0, 0, 0, 0, 1, 2, 4, 9, 18, 39, 80, 169, 350, 731, 1516, 3149, 6522, 13503, 27912, 57649, 118934, 245155, 504868, 1038869, 2135986, 4388487, 9009984, 18486009, 37904078, 77672299, 159072860, 325602269, 666117610, 1362061391, 2783775096, 5686854849, 11612318982
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO] (2012), p. 17 (Lemma 4.6).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).
FORMULA
G.f.: x^4*(1-3*x^2-x^3)/((1+x)*(1-2*x)*(1-x-2*x^2)).
a(n) = (2^(n-5)*(3*n+38)-(3*n-14)*(-1)^n)/27 with n>3, a(0)=a(1)=a(2)=a(3)=0. [Bruno Berselli, Nov 29 2012]
MATHEMATICA
CoefficientList[Series[x^4 (1 - 3 x^2 - x^3)/((1 + x) (1 - 2 x) (1 - x - 2 x^2)), {x, 0, 36}], x] (* Bruno Berselli, Nov 30 2012 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(1-3*x^2-x^3)/((1+x)*(1-2*x)*(1-x-2*x^2)), x, 0, n), x, n), n, 0, 36); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 1, 2, 4, 9]; [n le 8 select I[n] else 2*Self(n-1) + 3*Self(n-2) - 4*Self(n-3) - 4*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219756
Expansion of x^4*(1-7*x+17*x^2-18*x^3+11*x^4-5*x^5)/((1-x)^2*(1-3*x)^2*(1-3*x+x^2)^2).
+10
1
0, 0, 0, 0, 1, 7, 34, 141, 537, 1942, 6786, 23143, 77513, 256021, 836330, 2707652, 8701723, 27793375, 88310920, 279354069, 880300371, 2764788010, 8658249900, 27045078415, 84287831231, 262161737197, 813944768564, 2523027912296, 7809442203157, 24140652097687
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO] (2012), p. 18 (g.f. obtained from the Lemma 4.6).
Index entries for linear recurrences with constant coefficients, signature (14,-81,250,-444,458,-265,78,-9).
FORMULA
G.f.: x^4*(1-7*x+17*x^2-18*x^3+11*x^4-5*x^5)/((1-x)^2*(1-3*x)^2*(1-3*x+x^2)^2).
a(n) = 14*a(n-1) - 81*a(n-2) + 250*a(n-3) - 444*a(n-4) + 458*a(n-5) - 265*a(n-6) + 78*a(n-7) - 9*a(n-8). - Vincenzo Librandi, Dec 15 2012
MATHEMATICA
CoefficientList[Series[x^4 (1 - 7 x + 17 x^2 - 18 x^3 + 11 x^4 - 5 x^5)/((1 - x)^2 (1 - 3 x)^2 (1 - 3 x + x^2)^2), {x, 0, 29}], x] (* Bruno Berselli, Nov 30 2012 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(1-7*x+17*x^2-18*x^3+11*x^4-5*x^5)/((1-x)^2*(1-3*x)^2*(1-3*x+x^2)^2), x, 0, n), x, n), n, 0, 29); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 1, 7, 34, 141, 537, 1942, 6786, 23143]; [n le 12 select I[n] else 14*Self(n-1) - 81*Self(n-2) + 250*Self(n-3) - 444*Self(n-4) + 458*Self(n-5) -265*Self(n-6) + 78*Self(n-7) - 9*Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
A219757
Expansion of x^4*(2-7*x+6*x^2+x^3-x^4)/((1-x)*(1-2*x)^4*(1-3*x+x^2)).
+10
1
0, 0, 0, 0, 2, 17, 90, 383, 1436, 4958, 16159, 50480, 152690, 450343, 1301764, 3701990, 10387887, 28827688, 79265482, 216271927, 586261980, 1580524894, 4241295935, 11336890720, 30202962402, 80239307847, 212664541940, 562513804438, 1485379408591, 3916726647768
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO] (2012), p. 25.
Index entries for linear recurrences with constant coefficients, signature (12,-60,161,-248,216,-96,16).
FORMULA
G.f.: x^4*(2-7*x+6*x^2+x^3-x^4)/((1-x)*(1-2*x)^4*(1-3*x+x^2)).
MATHEMATICA
CoefficientList[Series[x^4(2-7x+6x^2+x^3-x^4)/((1-x)(1-2x)^4(1-3x+x^2)), {x, 0, 40}], x] (* Harvey P. Dale, Nov 29 2012 *)
PROG
(Maxima) makelist(coeff(taylor(x^4*(2-7*x+6*x^2+x^3-x^4)/((1-x)*(1-2*x)^4*(1-3*x+x^2)), x, 0, n), x, n), n, 0, 29); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 2, 17, 90, 383, 1436, 4958]; [n le 10 select I[n] else 12*Self(n-1) - 60*Self(n-2) + 161*Self(n-3) - 248*Self(n-4) + 216*Self(n-5) - 96*Self(n-6) + 16*Self(n-7): n in [1..30]]; // Vincenzo Librandi, Dec 14 2012
CROSSREFS
Cf. A219751-A219759, A219837.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 28 2012
STATUS
approved
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