OFFSET
1,2
COMMENTS
A related function is theta_3(x) = 1 + 2*Sum_{n>=1} 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} (x^n - 5*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 5*A(x))^(n-1) = 1 - 3*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 5*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 5*x^n*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 5*x^n*A(x))^(n+1) = 1 - 3*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 5*x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x + 5*x^2 + 24*x^3 + 111*x^4 + 506*x^5 + 2379*x^6 + 12083*x^7 + 67531*x^8 + 406284*x^9 + 2531203*x^10 + 15866775*x^11 + 98883303*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*x - 3*x^4 - 3*x^9 - 3*x^16 - 3*x^25 - 3*x^36 - 3*x^49 - ...
\SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05497127752043377386868704294930896868077597772598908285...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 5*A)^n = (5 - 3*Pi^(1/4)/gamma(3/4) )/2 = 0.870347783180037978...
(V.2) Let A = A(exp(-2*Pi)) = 0.001885037102906729934432374294398706956703235597857256076...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 5*A)^n = (5 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.994397671768391...
(V.3) Let A = A(-exp(-Pi)) = -0.03548990756971248576955893224372969073755967165800772531...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 5*A)^n = (5 - 3*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.12963129276582476...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001850160979277236538611428135062916090397865766804127684...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 5*A)^n = (5 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.005602328158639...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (x^m - 5*Ser(A))^m ) - 1 + 3*sum(m=1, #A, x^(m^2) ), #A-1)/5 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved