A370026 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

1, 6, 39, 269, 1917, 13893, 101830, 753255, 5614504, 42110432, 317474187, 2403893757, 18270065438, 139305459960, 1065183756535, 8165168139498, 62729216570805, 482878316552298, 3723769699813119, 28762830132956421, 222495155932381229, 1723432870654770161, 13366099075223254740

Offset

1,2

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..401

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^(n-1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^n = 0.

(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^(n+1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

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

Example

G.f.: A(x) = x + 6*x^2 + 39*x^3 + 269*x^4 + 1917*x^5 + 13893*x^6 + 101830*x^7 + 753255*x^8 + 5614504*x^9 + 42110432*x^10 + 317474187*x^11 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 - 8*x + 8*x^4 - 8*x^9 + 8*x^16 - 8*x^25 + 8*x^36 - 8*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05892551210473733684254468528377030200762221986684224912...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 6*A)^n = 4*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.65431655262446728562897...
(V.2) Let A = A(exp(-2*Pi)) = 0.001888624085511713374935799800784148455986111369097248489...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 6*A)^n = 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.985060458243628543159...
(V.3) Let A = A(-exp(-Pi)) = -0.03443859231795915470687740421610270983167641847531807729...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 6*A)^n = 4*Pi^(1/4)/gamma(3/4) - 3 = 1.3457392448532320583012...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001846769433141026637620872576636896819075507182864480219...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 6*A)^n = 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3 = 1.01493954195095636419...

Prog

(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 6*Ser(A))^m ) - 1 - 8*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/6 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370027, A370028, A370029, A370042.

Sequence in context: A052392 A370376 A357206 * A199491 A147961 A264232

Adjacent sequences: A370023 A370024 A370025 * A370027 A370028 A370029

Keyword

nonn

Author

Paul D. Hanna, Feb 09 2024

Status

approved