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.

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

Offset

0,2

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).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

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

Example

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)).

Prog

(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), ", "))

Crossrefs

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

Sequence in context: A361410 A229230 A135166 * A118347 A250887 A371577

Adjacent sequences: A363571 A363572 A363573 * A363575 A363576 A363577

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2023

Status

approved