A370031 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).

1, 1, 0, -1, 2, 15, 27, -1, -76, 19, 719, 1687, 184, -5976, -3749, 44093, 130933, 42026, -512833, -667101, 2976177, 11391169, 6608432, -45604863, -87819235, 202544340, 1053407806, 922859161, -4085924365, -10600384406, 12656739909, 100646660458, 121472828448, -360976456530

Offset: 1

Author

Paul D. Hanna, Feb 10 2024

Keywords

• sign

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = 1 + x - x^3 + 2*x^4 + 15*x^5 + 27*x^6 - x^7 - 76*x^8 + 19*x^9 + 719*x^10 + 1687*x^11 + 184*x^12 - 5976*x^13 - 3749*x^14 + 44093*x^15 + ...
where
Sum_{n=-oo..+oo} (x^n - A(x))^n = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04507828029039130528308497098432879536368681539259286273...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - A)^n = (1 + Pi^(1/4)/gamma(3/4))/2 = 1.0432174056066540...
(V.2) Let A = A(exp(-2*Pi)) = 0.001870930061948701432816606547007172908053584772650237678...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - A)^n = (1 + sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 1.00186744274386954...
(V.3) Let A = A(-exp(-Pi)) = -0.04135017416264159536574596265267969182735801577042441264...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - A)^n = (1 + (Pi/2)^(1/4)/gamma(3/4))/2 = 0.9567895690780584107...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001863955401558124515592555303127910405358304631205735085...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - A)^n = (1 + 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 0.99813255728045356...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..531
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A370041, A370030, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Programs

PARI

{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, (x^n - Ser(A))^n ) - sum(n=0,#A, x^(n^2) ), #A-1) ); A[n+1]}
for(n=1,40, print1(a(n),", "))

Formula

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

(1) Sum_{n=-oo..+oo} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).

(2) Sum_{n=-oo..+oo} x^n * (x^n + A(x))^(n-1) = Sum_{n>=0} x^(n^2).

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

(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*A(x))^n = Sum_{n>=0} x^(n^2).

(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^(n+1) = Sum_{n>=0} x^(n^2).

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