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URL: https://oeis.org/A370033

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A370033
G.f. A(x) satisfies Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
13
1, 3, 8, 19, 46, 161, 799, 4021, 17932, 71311, 268639, 1045731, 4464576, 20500010, 95221503, 429913365, 1879365529, 8112744634, 35452835755, 158833086233, 725458442577, 3329609464605, 15194309369384, 68837584452055, 311257278509193, 1413730859134250, 6469321177004978
OFFSET
1,2
COMMENTS
A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
LINKS
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 - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 3*A(x))^(n-1) = 1 - Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 3*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 3*x^n*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 3*x^n*A(x))^(n+1) = 1 - Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 3*x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 161*x^6 + 799*x^7 + 4021*x^8 + 17932*x^9 + 71311*x^10 + 268639*x^11 + 1045731*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 3*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)) = 0.04953636800560980886288845724196786482586224709976648461...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 3*A)^n = (3 - Pi^(1/4)/Gamma(3/4))/2 = 0.956782594393345992...
(V.2) Let A = A(exp(-2*Pi)) = 0.001877957090194880545086201853719041435355287864597005509...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 3*A)^n = (3 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/Gamma(3/4))/2 = 0.998132557256130454...
(V.3) Let A = A(-exp(-Pi)) = -0.03819699447470815952471171970837842342724818247967540335...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 3*A)^n = (3 - (Pi/2)^(1/4)/Gamma(3/4))/2 = 1.0432104309219415...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001857032573904813918259314464039219802478066024973444789...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 3*A)^n = (3 - 2^(1/8)*(Pi/2)^(1/4)/Gamma(3/4))/2 = 1.001867442719546432...
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 - 3*Ser(A))^m ) - 1 + sum(m=1, #A, x^(m^2) ), #A-1)/3 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved