A363574 - OEIS

%I #17 Nov 18 2023 07:58:10

%S 1,2,11,70,485,3586,27702,221044,1807751,15073208,127658948,

%T 1095160206,9496825919,83109648780,733063257227,6510317010502,

%U 58166005554886,522446273512866,4714846241261093,42730135199777198,388741207648594732,3548875263271057666,32500492203726887011

%N 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.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

%C Conjectures:

%C (1) [x^n/n] log(A(x)) == 0 (mod 2) for n >= 1,

%C (2) [x^n/n] log(A(x)) == 2 (mod 4) iff n is a square or twice a square (A028982).

%H Paul D. Hanna, <a href="/A363574/b363574.txt">Table of n, a(n) for n = 0..400</a>

%F 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).

%F (1) theta_4(x) = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).

%F (2) theta_4(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*A(x)*x^n)^(n+1).

%F (3) 2*A(x)*theta_4(x) = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - x^n)^(n-1).

%F (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).

%F (5) 0 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^n.

%F (6) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*x^n*A(x))^n.

%F a(n) ~ c * d^n / n^(3/2), where d = 9.7945249252767370556269070948885577825904333080336078... and c = 0.5596294216531106654141949766112236966734018523053... - _Vaclav Kotesovec_, Nov 18 2023

%e 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 + ...

%e By definition, theta_4(x) = P(x) + Q(x) where

%e theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36 - 2*x^49 + ...

%e 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) + ...

%e 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) + ...

%e Explicitly,

%e 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 + ...

%e 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 + ...

%e RELATED SERIES.

%e It appears that the coefficients of log(A(x)) are all even:

%e 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 + ...

%e SPECIFIC VALUES.

%e A(1/10) = 2.265719721251888941080447803329772146410479668...

%e A(-exp(-Pi)) = 0.92975039129846529364480115642201528102246496...

%e A(-exp(-2*Pi)) = 0.99630302525172375553562043431958560512563348...

%e A(exp(-Pi)) = 1.11512759518076350005641735660471754886478511...

%e where related values are

%e theta_4(-exp(-Pi)) = Pi^(1/4)/gamma(3/4),

%e theta_4(exp(-Pi)) = Pi^(1/4)/(gamma(3/4)*2^(1/4)).

%e For example, we have

%e 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...

%e also,

%e 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)).

%o (PARI) {theta_4(m) = sum(n=-sqrtint(m+1),sqrtint(m+1), (-1)^n * x^(n^2) + x*O(x^m))}

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

%o 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]}

%o for(n=0, 30, print1(a(n), ", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 10 2023