|
%I #13 Nov 18 2023 07:58:21
%S 2,18,152,1298,11432,102528,931968,8554698,79116722,736053548,
%T 6880854312,64581401504,608178222316,5743797385400,54380418647072,
%U 515969530624186,4904912090029220,46705792266725778,445414424602046360,4253490292103734268,40668438064085218644,389270517755459345232
%N Expansion of l.g.f. A(x) satisfying theta_4(x) = Sum_{n=-oo..+oo} x^n * (2*exp(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 Conjectures:
%C (1) a(n) == 0 (mod 2) for n >= 1,
%C (2) a(n) == 2 (mod 4) iff n is a square or twice a square (A028982) and n > 0.
%H Paul D. Hanna, <a href="/A363568/b363568.txt">Table of n, a(n) for n = 1..400</a>
%F L.g.f. A(x) = Sum_{n>=1} a(n)*x^n/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*exp(A(x)) - x^n)^(n-1).
%F (2) theta_4(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*exp(A(x))*x^n)^(n+1).
%F (3) 2*exp(A(x))*theta_4(x) = Sum_{n=-oo..+oo} x^(2*n) * (2*exp(A(x)) - x^n)^(n-1).
%F (4) 2*exp(A(x))*theta_4(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*exp(A(x))*x^n)^(n+1).
%F a(n) ~ c * d^n / sqrt(n), where d = 9.794524925276737055626907094888557782590433308033607879... and c = 0.289888114090505739891516837512006241321867679200012... - _Vaclav Kotesovec_, Nov 18 2023
%e L.g.f.: 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 + ...
%e where exponentiation yields the g.f. of A363574, which begins
%e exp(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 + ... + A363574(n)*x^n + ...
%e and
%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 SPECIFIC VALUES.
%e A(1/10) = 0.8178924661878061760306420027592911481425...
%e A(-exp(-Pi)) = -0.0728391253128727098719433234114815730792...
%e A(-exp(-2*Pi)) = -0.0037038254492119550196758540871566838036...
%e A(exp(-Pi)) = 0.108968833504579170457547429339378954140155...
%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*exp(A(exp(-Pi))) - exp(-n*Pi))^(n-1) = Pi^(1/4)/(gamma(3/4)*2^(1/4)) = 0.91357913815611682...
%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); n*polcoeff(log(Ser(A)),n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A363574 (exp(A(x))), A002448 (theta_4).
%K nonn
%O 1,1
%A _Paul D. Hanna_, Jun 11 2023
| 