A350483 -id:A350483

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A350475

G.f. A(x) satisfies: A(x) = A(x^2 - x^4)/x.

+10
5

1, -1, -1, 3, -4, 6, -7, -11, 54, -68, -33, 207, -160, -398, 1087, -461, -3370, 11120, -18761, 9287, 52861, -191783, 325226, -164758, -710654, 2437586, -4100514, 2865198, 6070531, -26403315, 51585667, -50596137, -38932192, 280507198

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

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..34.

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:

(1) A(x) = A(x^2 - x^4)/x.

(2) R(x*A(x)) = x^2 - x^4, where R(A(x)) = x.

(3) A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x^2, and F(n+1) = 1 - (1 - F(n))^2 * F(n)^2 for n > 0.

EXAMPLE

G.f.: A(x) = x - x^3 - x^5 + 3*x^7 - 4*x^9 + 6*x^11 - 7*x^13 - 11*x^15 + 54*x^17 - 68*x^19 - 33*x^21 + 207*x^23 - 160*x^25 + ...

The series reversion is here denoted R(x) so that R(A(x)) = x where

R(x) = x + x^3 + 4*x^5 + 17*x^7 + 89*x^9 + 487*x^11 + 2835*x^13 + 17039*x^15 + 105390*x^17 + ... + A350474(n)*x^(2*n-1) + ...

and which by definition also satisfies R(x*A(x)) = x^2 - x^4.

GENERATING METHOD.

One may generate the g.f. A(x) using the following method.

Define F(n), a polynomial in x of order 2^(2*n-1), by the following recurrence:

F(0) = x,

F(1) = (1 - x^2),

F(2) = (1 - x^4 * (1-x^2)^2),

F(3) = (1 - x^8 * (1-x^2)^4 * F(2)^2),

F(4) = (1 - x^16 * (1-x^2)^8 * F(2)^4 * F(3)^2),

F(5) = (1 - x^32 * (1-x^2)^16 * F(2)^8 * F(3)^4 * F(4)^2),

...

F(n+1) = 1 - (1 - F(n))^2 * F(n)^2

...

Then the g.f. A(x) equals the infinite product:

A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

that is,

A(x) = x * (1-x^2) * (1 - x^4*(1-x^2)^2) * (1 - x^8*(1-x^2)^4*(1 - x^2*(1-x^2)^2)^2) * (1 - x^16*(1-x^2)^8*(1 - x^2*(1-x^2)^2)^4*(1 - x^4*(1-x^2)^4*(1 - x^2*(1-x^2)^2)^2)^2) * ...

SPECIFIC VALUES.

The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.

A(1/2) = (1/2) * (3/2^2) * (247/2^8) * (4290025567/2^32) * ... = 0.36139962803961485982912810261...

A(2/3) = (2/3) * (5/3^2) * (6161/3^8) * (1846946921491841/3^32) * ... = 0.34664667253812499241960389380...

A(1/3) = (1/3) * (8/3^2) * (6497/3^8) * (1852847292558977/3^32) * ... = 0.29337866246734888661663159512...

The first relative maximum value of A(x) is given by

A(0.554161678235...) = 0.365908834442...

PROG

(PARI) {a(n) = my(A, R=[1, 0]); for(i=1, n, R=concat(R, 0);

R[#R] = -polcoeff( x^2*(1 - x^2) - subst(x*Ser(R), x, x * serreverse(x*Ser(R))), #R+1) );

A=Vec(serreverse(x*Ser(R))); A[n]}

for(n=1, 40, print1(a(2*n-1), ", "))

(PARI) /* Using Infinite Product Formula for Series Reversion */

{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^2), G = 1 - (1 - F(n-1))^2 * F(n-1)^2 )); G}

{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}

for(n=1, 40, print1(a(2*n-1), ", "))

CROSSREFS

Cf. A350474 (inverse), A273218, A350431, A350433, A350477, A350479, A350481, A350483.

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 01 2022

STATUS

approved

A350477

G.f. A(x) satisfies: A(x) = A(x^2 - x^5)/x.

+10
5

1, -1, -1, 4, -7, 11, -18, -5, 138, -368, 298, 819, -2863, 2711, 5105, -18524, 8298, 92719, -344351, 606264, -38650, -3668127, 13802755, -27695500, 16311302, 102383087, -440646648, 939806924, -847930096, -2006926300, 11102911188, -27232242899, 35448779213

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

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..33.

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) satisfies:

(1) A(x) = A(x^2 - x^5)/x.

(2) R(x*A(x)) = x^2 - x^5, where R(A(x)) = x.

(3) A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x^3, and F(n+1) = 1 - (1 - F(n))^2 * F(n)^3 for n > 0.

EXAMPLE

G.f.: A(x) = x - x^4 - x^7 + 4*x^10 - 7*x^13 + 11*x^16 - 18*x^19 - 5*x^22 + 138*x^25 - 368*x^28 + 298*x^31 + 819*x^34 + ...

The series reversion is here denoted R(x) so that R(A(x)) = x where

R(x) = x + x^4 + 5*x^7 + 29*x^10 + 203*x^13 + 1519*x^16 + 12047*x^19 + 99112*x^22 + ... + A350476(n)*x^(3*n-2) + ...

and which by definition also satisfies R(x*A(x)) = x^2 - x^5.

GENERATING METHOD.

One may generate the g.f. A(x) using the following method.

Define F(n), a polynomial in x of order 3*5^(n-1), by the following recurrence:

F(0) = x,

F(1) = (1 - x^3),

F(2) = (1 - x^6 * (1-x^3)^3),

F(3) = (1 - x^12 * (1-x^3)^6 * F(2)^3),

F(4) = (1 - x^24 * (1-x^3)^12 * F(2)^6 * F(3)^3),

F(5) = (1 - x^48 * (1-x^3)^24 * F(2)^12 * F(3)^6 * F(4)^3),

...

F(n+1) = 1 - (1 - F(n))^2 * F(n)^3

...

Then the g.f. A(x) equals the infinite product:

A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

that is,

A(x) = x * (1-x^3) * (1 - x^6*(1-x^3)^3) * (1 - x^12*(1-x^3)^6*(1 - x^6*(1-x^3)^3)^3) * (1 - x^24*(1-x^3)^12*(1 - x^6*(1-x^3)^3)^6*(1 - x^12*(1-x^3)^6*(1 - x^6*(1-x^3)^3)^3)^3) * ...

SPECIFIC VALUES.

The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.

A(1/2) = (1/2) * (7/2^3) * (32425/2^15) * (37774921088925702693943/2^75) * ... = 0.43287449035613254837986255178...

A(2/3) = (2/3) * (19/3^3) * (13909931/3^15) * (607748159091514279939881665134154491/3^75) * ... = 0.45439544136368649551815752865...

A(1/3) = (1/3) * (26/3^3) * (14331331/3^15) * (608265878429109169424505294701548091/3^75) * ... = 0.32059399676030866078416260973...

The first relative maximum value of A(x) is given by

A(0.6158833582739456...) = 0.46015216135279854...

PROG

(PARI) {a(n) = my(A, R=[1, 0]); for(i=1, n, R=concat(R, 0);

R[#R] = -polcoeff( x^2*(1 - x^3) - subst(x*Ser(R), x, x * serreverse(x*Ser(R))), #R+1) );

A=Vec(serreverse(x*Ser(R))); A[n]}

for(n=1, 40, print1(a(3*n-2), ", "))

(PARI) /* Using Infinite Product Formula for Series Reversion */

N = 300; \\ set limit on order of polynomials to be 3 times desired number of terms

{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = 1-x^3, G = 1 - (1 - F(n-1))^2 * F(n-1)^3 +x^2*O(x^N) )); G}

{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}

for(n=1, 40, print1(a(3*n-2), ", "))

CROSSREFS

Cf. A350476 (inverse), A273218, A350431, A350433, A350475, A350479, A350481, A350483.

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 01 2022

STATUS

approved

A350481

G.f. A(x) satisfies: A(x) = A(x^3 - x^6)/x^2.

+10
5

1, -1, 0, -1, 4, -6, 4, -1, 0, -1, 10, -45, 124, -262, 564, -1360, 3076, -5913, 10238, -17861, 32040, -54785, 85310, -124275, 178760, -256919, 354870, -455046, 543444, -623196, 702354, -775965, 854874, -1083885, 2001516, -5152914, 14709734

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

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..37.

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) satisfies:

(1) A(x) = A(x^3 - x^6)/x^2.

(2) R(x^2*A(x)) = x^3 - x^6, where R(A(x)) = x.

(3) A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x^3, and F(n+1) = 1 - (1 - F(n))^3 * F(n)^3 for n > 0.

EXAMPLE

G.f.: A(x) = x - x^4 - x^10 + 4*x^13 - 6*x^16 + 4*x^19 - x^22 - x^28 + 10*x^31 - 45*x^34 + 124*x^37 - 262*x^40 + ...

The series reversion is here denoted R(x) so that R(A(x)) = x where

R(x) = x + x^4 + 4*x^7 + 23*x^10 + 150*x^13 + 1060*x^16 + 7910*x^19 + 61319*x^22 + 488982*x^25 + ... + A350480(n)*x^(3*n-2) + ...

and which by definition also satisfies R(x^2*A(x)) = x^3 - x^6.

GENERATING METHOD.

One may generate the g.f. A(x) using the following method.

Define F(n), a polynomial in x of order 3*6^(n-1), by the following recurrence:

F(0) = x,

F(1) = (1 - x^3),

F(2) = (1 - x^9 * (1-x^3)^3),

F(3) = (1 - x^27 * (1-x^3)^9 * F(2)^3),

F(4) = (1 - x^81 * (1-x^3)^27 * F(2)^9 * F(3)^3),

F(5) = (1 - x^243 * (1-x^3)^81 * F(2)^27 * F(3)^9 * F(4)^3),

...

F(n+1) = 1 - (1 - F(n))^3 * F(n)^3

...

Then the g.f. A(x) equals the infinite product:

A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

that is,

A(x) = x * (1-x^3) * (1 - x^9*(1-x^3)^3) * (1 - x^27*(1-x^3)^9*(1 - x^9*(1-x^3)^3)^3) * (1 - x^81*(1-x^3)^27*(1 - x^9*(1-x^3)^3)^9*(1 - x^27*(1-x^3)^9*(1 - x^9*(1-x^3)^3)^3)^3) * ...

SPECIFIC VALUES.

The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.

A(1/2) = (1/2) * (7/2^3) * (261801/2^18) * (324518552934330535025670717244849/2^108) * ... = 0.43692755601666346006790651129...

A(2/3) = (2/3) * (19/3^3) * (383908681/3^18) * ... = 0.46488294171447073294042484530...

A(1/3) = (1/3) * (26/3^3) * (387402913/3^18) * ... = 0.32097309216130808472175568906...

The first relative maximum value of A(x) is given by

A(0.62382241029098620649978...) = 0.46944663707521689313693...

PROG

(PARI) {a(n) = my(A, R=[1, 0]); for(i=1, n, R=concat(R, 0);

R[#R] = -polcoeff( x^3*(1 - x^3) - subst(x*Ser(R), x, x^2 * serreverse(x*Ser(R))), #R+2) );

A=Vec(serreverse(x*Ser(R))); H=A; A[n]}

for(n=1, 40, print1(a(3*n-2), ", "))

(PARI) /* Using Infinite Product Formula for Series Reversion */

N = 300; \\ set limit on order of polynomials to be 3 times desired number of terms

{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^3), G = 1 - (1 - F(n-1))^3 * F(n-1)^3 +x^2*O(x^N) )); G}

{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}

for(n=1, 40, print1(a(3*n-2), ", "))

CROSSREFS

Cf. A350480 (inverse), A273218, A350431, A350433, A350475, A350477, A350483.

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 01 2022

STATUS

approved

A350482

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies: A(x^3*R(x)) = x^4 - x^6, where A(R(x)) = x.

+10
5

1, 1, 3, 12, 56, 282, 1494, 8207, 46332, 267174, 1566994, 9318630, 56058288, 340530734, 2085902781, 12869679276, 79906734738, 498903972318, 3130391261901, 19728829226876, 124833794989761, 792731281119894, 5050567538514688, 32273851864657050, 206799487942332132

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..25.

FORMULA

G.f.: A(x) = Series_Reversion( Product_{n>=0} F(n) ), where F(0) = x, F(1) = 1-x^2, and F(n+1) = 1 - (1 - F(n))^4 * F(n)^2 for n > 0.

EXAMPLE

G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 56*x^9 + 282*x^11 + 1494*x^13 + 8207*x^15 + 46332*x^17 + 267174*x^19 + ...

The series reversion is here denoted R(x) so that R(A(x)) = x where

R(x) = x - x^3 - x^9 + 3*x^11 - 3*x^13 + x^15 - x^33 + 9*x^35 - 36*x^37 + 84*x^39 + ... + A350483(n)*x^(2*n-1) + ...

and which by definition also satisfies A(x^3*R(x)) = x^4 - x^6.

GENERATING METHOD.

One may obtain the g.f. A(x) from the following method used to generate the series reversion R(x).

Define F(n), a polynomial in x of order 2*6^(n-1), by the following recurrence:

F(0) = x,

F(1) = (1 - x^2),

F(2) = (1 - x^8 * (1-x^2)^2),

F(3) = (1 - x^32 * (1-x^2)^8 * F(2)^2),

F(4) = (1 - x^128 * (1-x^2)^32 * F(2)^8 * F(3)^2),

F(5) = (1 - x^512 * (1-x^2)^128 * F(2)^32 * F(3)^8 * F(4)^2),

...

F(n+1) = 1 - (1 - F(n))^4 * F(n)^2

...

Then the series reversion R(x) equals the infinite product:

R(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

that is,

R(x) = x * (1-x^2) * (1 - x^8*(1-x^2)^2) * (1 - x^32*(1-x^2)^8*(1 - x^8*(1-x^2)^2)^2) * (1 - x^128*(1-x^2)^32*(1 - x^8*(1-x^2)^2)^8*(1 - x^32*(1-x^2)^8*(1 - x^8*(1-x^2)^2)^2)^2) * ...

The g.f. of this sequence is then obtained as the series reversion of this infinite product.

PROG

(PARI) {a(n) = my(A=[1, 0]); for(i=1, n, A=concat(A, 0);

A[#A] = -polcoeff( x^4*(1 - x^2) - subst(x*Ser(A), x, x^3 * serreverse(x*Ser(A))), #A+3) ); A[n]}

for(n=1, 30, print1(a(2*n-1), ", "))

(PARI) /* Using Infinite Product Formula for Series Reversion */

N = 300; \\ set limit on order of polynomials to be 2 times desired number of terms

{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^2), G = 1 - (1 - F(n-1))^4 * F(n-1)^2 +x*O(x^N) )); G}

{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(serreverse(A), n)}

for(n=1, 30, print1(a(2*n-1), ", "))

CROSSREFS

Cf. A350483 (inverse), A273162, A273203, A350434, A350474, A350476, A350478, A350480.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 01 2022

STATUS

approved

A350479

G.f. A(x) satisfies: A(x) = A(x^3 - x^5)/x^2.

+10
4

1, -1, 0, -1, 3, -3, 1, 0, 0, -1, 7, -21, 38, -62, 129, -262, 412, -543, 748, -1111, 1491, -1675, 1705, -1782, 1881, -1749, 1290, -716, 305, -249, 989, -4024, 13098, -36463, 92766, -223338, 509455, -1093243, 2218041, -4315247, 8126815, -14817936, 26133655

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

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..43.

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:

(1) A(x) = A(x^3 - x^5)/x^2.

(2) R(x^2*A(x)) = x^3 - x^5, where R(A(x)) = x.

(3) A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x^2, and F(n+1) = 1 - (1 - F(n))^3 * F(n)^2 for n > 0.

EXAMPLE

G.f.: A(x) = x - x^3 - x^7 + 3*x^9 - 3*x^11 + x^13 - x^19 + 7*x^21 - 21*x^23 + 38*x^25 - 62*x^27 + 129*x^29 - 262*x^31 + ...

The series reversion is here denoted R(x) so that R(A(x)) = x where

R(x) = x + x^3 + 3*x^5 + 13*x^7 + 62*x^9 + 318*x^11 + 1721*x^13 + 9660*x^15 + 55710*x^17 + ... + A350478(n)*x^(2*n-1) + ...

and which by definition also satisfies R(x^2*A(x)) = x^3 - x^5.

GENERATING METHOD.

One may generate the g.f. A(x) using the following method.

Define F(n), a polynomial in x of order 2*5^(n-1), by the following recurrence:

F(0) = x,

F(1) = (1 - x^2),

F(2) = (1 - x^6 * (1-x^2)^2),

F(3) = (1 - x^18 * (1-x^2)^6 * F(2)^2),

F(4) = (1 - x^54 * (1-x^2)^18 * F(2)^6 * F(3)^2),

F(5) = (1 - x^162 * (1-x^2)^54 * F(2)^18 * F(3)^6 * F(4)^2),

...

F(n+1) = 1 - (1 - F(n))^3 * F(n)^2

...

Then the g.f. A(x) equals the infinite product:

A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

that is,

A(x) = x * (1-x^2) * (1 - x^6*(1-x^2)^2) * (1 - x^18*(1-x^2)^6*(1 - x^6*(1-x^2)^2)^2) * (1 - x^54*(1-x^2)^18*(1 - x^6*(1-x^2)^2)^6*(1 - x^18*(1-x^2)^6*(1 - x^6*(1-x^2)^2)^2)^2) * ...

SPECIFIC VALUES.

The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.

A(1/2) = (1/2) * (3/2^2) * (1015/2^10) * (1125899155808599/2^50) * ... = 0.37170385361645629840998279262...

A(2/3) = (2/3) * (5/3^2) * (57449/3^10) * (717897986779793260667849/3^50) * ... = 0.36032797749163984225405820293...

A(1/3) = (1/3) * (8/3^2) * (58985/3^10) * (717897986779793260667849/3^50) * ... = 0.29597515647738646568618474726...

The first relative maximum value of A(x) is given by

A(0.5648072584544076680095600...) = 0.3788377227391210352270204...

PROG

(PARI) {a(n) = my(A, R=[1, 0]); for(i=1, n, R=concat(R, 0);

R[#R] = -polcoeff( x^3*(1 - x^2) - subst(x*Ser(R), x, x^2 * serreverse(x*Ser(R))), #R+2) );

A=Vec(serreverse(x*Ser(R))); A[n]}

for(n=1, 60, print1(a(2*n-1), ", "))

(PARI) /* Using Infinite Product Formula */

N = 300; \\ set limit on order of polynomials to be 2 times desired number of terms

{F(n) = my(G=x); if(n==0, G=x, if(n==1, G = (1-x^2), G = 1 - (1 - F(n-1))^3 * F(n-1)^2 +x^2*O(x^N) )); G}

{a(n) = my(A = prod(k=0, #binary(n), F(k) +x*O(x^n))); polcoeff(A, n)}

for(n=1, 60, print1(a(2*n-1), ", "))

CROSSREFS

Cf. A350478 (inverse), A273218, A350431, A350433, A350475, A350477, A350481, A350483.

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 01 2022

STATUS

approved

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