cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A363112 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).

Original entry on oeis.org

1, 1, 6, 51, 470, 4716, 49350, 534115, 5929892, 67175779, 773473709, 9025907984, 106511693025, 1268898400188, 15240421643846, 184348620664449, 2243749948233175, 27459089491691552, 337685454820968084, 4170918486201555250, 51719670553572755173, 643610071084847351183
Offset: 0

Views

Author

Paul D. Hanna, May 14 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 51*x^3 + 470*x^4 + 4716*x^5 + 49350*x^6 + 534115*x^7 + 5929892*x^8 + 67175779*x^9 + 773473709*x^10 + ...

Links

Crossrefs

Cf. A357227, A363113, A363114.
Cf. A361772.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(2*m-1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, x^(2*m^2)/(1 - 2*Ser(A)*x^m)^(2*m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).
(2) -1 = Sum_{n=-oo..+oo} x^(2*n^2) / (1 - 2*A(x)*x^n)^(2*n+1).

A363113 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(3*n-1).

Original entry on oeis.org

1, 2, 30, 621, 14196, 351802, 9179386, 248533626, 6917835992, 196730606200, 5691264122213, 166961281712818, 4955321842136163, 148522859439511133, 4489164688548477495, 136677755757518772050, 4187859771944659634378, 129039023692329781903247, 3995878021838502688832856
Offset: 0

Views

Author

Paul D. Hanna, May 14 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 30*x^2 + 621*x^3 + 14196*x^4 + 351802*x^5 + 9179386*x^6 + 248533626*x^7 + 6917835992*x^8 + 196730606200*x^9 + ...

Links

Crossrefs

Cf. A357227, A363112, A363114.
Cf. A361773.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(3*m-1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-1 + sum(m=-#A, #A, (-1)^(m+1) * x^(3*m^2)/(1 - 2*Ser(A)*x^m)^(3*m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(3*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n^2) / (1 - 2*A(x)*x^n)^(3*n+1).

A361778 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.

Original entry on oeis.org

1, 2, 7, 27, 109, 459, 2006, 9017, 41384, 193048, 912571, 4361939, 21045710, 102361864, 501349447, 2470556294, 12240270901, 60935582862, 304660949343, 1529125824203, 7701783889261, 38915600049447, 197206343307012, 1002023916642621, 5103911800972155, 26056404563941575
Offset: 0

Views

Author

Paul D. Hanna, May 10 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 27*x^3 + 109*x^4 + 459*x^5 + 2006*x^6 + 9017*x^7 + 41384*x^8 + 193048*x^9 + 912571*x^10 + ...
SPECIFIC VALUES.
A(1/7) = 1.63053651133635034184414884744745628155427916612173429157...
A(1/6) = 1.99892384479086071017436459041327119822244448085100733509...
A(x) = 2 at x = 0.166713109990638926829644490786806133084979604287174064...
Radius of convergence r and the value A(r) are given by
r = 0.182033752413024354859591633469061831146023401652842514076551...
A(r) = 2.63999965897091399750291467200041973752650665197493948118984006...
1/r = 5.4934867119096473651972990947886642212447897087082048838...

Links

Crossrefs

Cf. A355865, A357227, A359712.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, x^m * ((-x)^(m-1) - 2*Ser(A))^m ), #A)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (2*A(x) - (-x)^n)^n.
(3) 2*A(x) = Sum_{n=-oo..+oo} x^(3*n+1) * ((-x)^n - 2*A(x))^n.
(4) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^n.
(5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
(6) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
(7) 0 = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - (-x)^n)^(n+1).
(8) 0 = Sum_{n=-oo..+oo} x^(3*n) * ((-x)^(n-1) - 2*A(x))^(n+1).

A363114 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(4*n-1).

Original entry on oeis.org

1, 4, 138, 6571, 353935, 20694945, 1276853497, 81834405039, 5395444806588, 363600236084796, 24933767742193052, 1734273108108910743, 122058422998192278797, 8676376795137864622232, 622018188741046650309066, 44922343315319150402783783, 3265215115112327274815579250
Offset: 0

Views

Author

Paul D. Hanna, May 14 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 138*x^2 + 6571*x^3 + 353935*x^4 + 20694945*x^5 + 1276853497*x^6 + 81834405039*x^7 + 5395444806588*x^8 + ...

Crossrefs

Cf. A357227, A363112, A363113.
Cf. A361774.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(4*m-1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, x^(4*m^2)/(1 - 2*Ser(A)*x^m)^(4*m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(4*n-1).
(2) -1 = Sum_{n=-oo..+oo} x^(4*n^2) / (1 - 2*A(x)*x^n)^(4*n+1).

A363574 Expansion of g.f. A(x) satisfying theta_4(x) = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1) where theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2) is a Jacobi theta function.

Original entry on oeis.org

1, 2, 11, 70, 485, 3586, 27702, 221044, 1807751, 15073208, 127658948, 1095160206, 9496825919, 83109648780, 733063257227, 6510317010502, 58166005554886, 522446273512866, 4714846241261093, 42730135199777198, 388741207648594732, 3548875263271057666, 32500492203726887011
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
Conjectures:
(1) [x^n/n] log(A(x)) == 0 (mod 2) for n >= 1,
(2) [x^n/n] log(A(x)) == 2 (mod 4) iff n is a square or twice a square (A028982).

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 70*x^3 + 485*x^4 + 3586*x^5 + 27702*x^6 + 221044*x^7 + 1807751*x^8 + 15073208*x^9 + 127658948*x^10 + ...
By definition, theta_4(x) = P(x) + Q(x) where
theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36 - 2*x^49 + ...
P(x) = x + x^2*(2*A(x) - x^2) + x^3*(2*A(x) - x^3)^2 + x^4*(2*A(x) - x^4)^3 + x^5*(2*A(x) - x^5)^4 + x^6*(2*A(x) - x^6)^5 + ... + x^n*(2*A(x) - x^n)^(n-1) + ...
Q(x) = 1/(2*A(x) - 1) + x/(1 - 2*A(x)*x)^2 - x^4/(1 - 2*A(x)*x^2)^3 + x^9/(1 - 2*A(x)*x^3)^4 - x^16/(1 - 2*A(x)*x^4)^5 + ... + (-1)^(n+1)*x^(n^2)/(1 - 2*A(x)*x^n)^(n+1) + ...
Explicitly,
P(x) = x + 2*x^2 + 8*x^3 + 45*x^4 + 308*x^5 + 2222*x^6 + 16920*x^7 + 133428*x^8 + 1081337*x^9 + 8950618*x^10 + ...
Q(x) = 1 - 3*x - 2*x^2 - 8*x^3 - 43*x^4 - 308*x^5 - 2222*x^6 - 16920*x^7 - 133428*x^8 - 1081339*x^9 + ...
RELATED SERIES.
It appears that the coefficients of log(A(x)) are all even:
log(A(x)) = 2*x + 18*x^2/2 + 152*x^3/3 + 1298*x^4/4 + 11432*x^5/5 + 102528*x^6/6 + 931968*x^7/7 + 8554698*x^8/8 + 79116722*x^9/9 + ... + A363568(n)*x^n/n + ...
SPECIFIC VALUES.
A(1/10) = 2.265719721251888941080447803329772146410479668...
A(-exp(-Pi)) = 0.92975039129846529364480115642201528102246496...
A(-exp(-2*Pi)) = 0.99630302525172375553562043431958560512563348...
A(exp(-Pi)) = 1.11512759518076350005641735660471754886478511...
where related values are
theta_4(-exp(-Pi)) = Pi^(1/4)/gamma(3/4),
theta_4(exp(-Pi)) = Pi^(1/4)/(gamma(3/4)*2^(1/4)).
For example, we have
Sum_{n=-oo..+oo} exp(-n*Pi) * (2*A(exp(-Pi)) - exp(-n*Pi))^(n-1) = Pi^(1/4)/(gamma(3/4)*2^(1/4)) = 0.91357913815611682...
also,
Sum_{n=-oo..+oo} (-1)^(n+1) * exp(-n^2*Pi) / (1 - 2*A(exp(-Pi))*exp(-n*Pi))^(n+1) = Pi^(1/4)/(gamma(3/4)*2^(1/4)).

Links

Crossrefs

Cf. A363568 (log(A(x))), A357227, A002448 (theta_4), A028982.

Programs

  • PARI
    {theta_4(m) = sum(n=-sqrtint(m+1),sqrtint(m+1), (-1)^n * x^(n^2) + x*O(x^m))}
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-theta_4(#A) + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(m-1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas; here theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(1) theta_4(x) = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).
(2) theta_4(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*A(x)*x^n)^(n+1).
(3) 2*A(x)*theta_4(x) = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - x^n)^(n-1).
(4) 2*A(x)*theta_4(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*A(x)*x^n)^(n+1).
(5) 0 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*x^n*A(x))^n.
a(n) ~ c * d^n / n^(3/2), where d = 9.7945249252767370556269070948885577825904333080336078... and c = 0.5596294216531106654141949766112236966734018523053... - Vaclav Kotesovec, Nov 18 2023
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