A370023 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).

1, 3, 12, 53, 234, 1041, 4711, 21573, 99484, 461657, 2154591, 10102701, 47555840, 224624016, 1064183887, 5055060411, 24068888061, 114841741098, 548992775523, 2628924592737, 12608597616161, 60558351876803, 291238387762452, 1402314223189959, 6759651098793285, 32617445956236720

Offset: 1

Author

Paul D. Hanna, Feb 09 2024

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = x + 3*x^2 + 12*x^3 + 53*x^4 + 234*x^5 + 1041*x^6 + 4711*x^7 + 21573*x^8 + 99484*x^9 + 461657*x^10 + 2154591*x^11 + 10102701*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 - 5*x + 5*x^4 - 5*x^9 + 5*x^16 - 5*x^25 + 5*x^36 - 5*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05001316702398359971645418498866690386932728399152644693...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 3*A)^n = (5*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.78394784539029205351810...
(V.2) Let A = A(exp(-2*Pi)) = 0.001877983557643657576778844718492775838546798118866577860...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 3*A)^n = (5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.990662786402267839474...
(V.3) Let A = A(-exp(-Pi)) = -0.03842474691590612761867206263978602696713545771404819339...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 3*A)^n = (5*Pi^(1/4)/gamma(3/4) - 3)/2 = 1.216087028033270036438...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001857058214293085256892081751882664927312970576990961749...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 3*A)^n = (5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3)/2 = 1.009337213719347727619...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..401
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A370020, A370021, A370022, A370024, A370025, A370026, A370027, A370028, A370029, A370042.

Programs

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 + 3*Ser(A))^m ) - 1 - 5*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/3 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

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 + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).

(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^(n-1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).

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

(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).

(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^(n+1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).

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