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..326
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 - 9*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 9*A(x))^(n-1) = 1 - 7*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 9*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 9*x^n*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 9*x^n*A(x))^(n+1) = 1 - 7*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 9*x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x + 9*x^2 + 80*x^3 + 703*x^4 + 6130*x^5 + 53351*x^6 + 466315*x^7 + 4118167*x^8 + 36941188*x^9 + 337853203*x^10 + 3155619199*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 7*x - 7*x^4 - 7*x^9 - 7*x^16 - 7*x^25 - 7*x^36 - 7*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07041342765468695859173243504212855904085321490660808668...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 9*A)^n = (9 - 7*Pi^(1/4)/gamma(3/4))/2 = 0.69747816075342194898639...
(V.2) Let A = A(exp(-2*Pi)) = 0.001899358496977867055016493259704554658290299283307899768...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 9*A)^n = (9 - 7*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.98692790079291318133312...
(V.3) Let A = A(-exp(-Pi)) = -0.03108273985731889208644710399967055047528520340415555251...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 9*A)^n = (9 - 7*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.302473016453591125074...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001836569230890760040434767580223720991124539653197115902...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 9*A)^n = (9 - 7*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.013072099036825024735...
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 - 9*Ser(A))^m ) - 1 + 7*sum(m=1, #A, x^(m^2) ), #A-1)/9 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved