A370042 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, 178717112, 2020360748, 22947819571, 261696375829, 2994717484790, 34373295184712, 395580223408591, 4563146810297938, 52747346257279381, 610871638149166758, 7086520419499114527, 82334442066436896541, 957935578573905521101

Offset: 1

Author

Paul D. Hanna, Feb 08 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 + 10*x^2 + 103*x^3 + 1089*x^4 + 11749*x^5 + 128637*x^6 + 1423982*x^7 + 15898231*x^8 + 178717112*x^9 + 2020360748*x^10 + 22947819571*x^11 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 - 12*x + 12*x^4 - 12*x^9 + 12*x^16 - 12*x^25 + 12*x^36 - 12*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07738488286995169642543180751945321776018365032150702566...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 10*A)^n = 6*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.4814748289367009284...
(V.2) Let A = A(exp(-2*Pi)) = 0.001903000450057888437867399675031908155434474357834107336...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 10*A)^n = 6*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.9775906873654428147...
(V.3) Let A = A(-exp(-Pi)) = -0.03025721520362353256298796517975081121112509387406260314...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 10*A)^n = 6*Pi^(1/4)/gamma(3/4) - 5 = 1.518608867279848087...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001833227104147111248063467420834469150941590720555554025...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 10*A)^n = 6*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5 = 1.0224093129264345...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..272
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029.

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 + 10*Ser(A))^m ) - 1 - 12*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/10 ); 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 + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^(n-1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^n = 0.

(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^(n+1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 10*A(x)*x^n)^(n+1) = 0.