OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
FORMULA
G.f. satisfies: A(x) = Product_{n>=1} (1 - x^(2n))*(1 + x^(2n-1)*A(x))*(1 + x^(2n-1)/A(x)), due to the Jacobi triple product identity.
a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 4.73097028144959... and c = 0.1236197969613... . - Vaclav Kotesovec, Mar 02 2016
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^3 - 6*x^4 + 32*x^5 - 88*x^6 + 376*x^7 +...
The g.f. A(x) satisfies the series:
* A(x) = 1 + (A(x) + A(x)^-1)*x + (A(x)^2 + A(x)^-2)*x^4 + (A(x)^3 + A(x)^-3)*x^9 + (A(x)^4 + A(x)^-4)*x^16 +...
* A(x) = (1-x^2)*(1+x*A(x))*(1+x/A(x)) * (1-x^4)*(1+x^3*A(x))*(1+x^3/A(x)) * (1-x^6)*(1+x^5*A(x))*(1+x^5/A(x)) *...
which is a result due to the Jacobi triple product identity.
MATHEMATICA
(* Calculation of constant d: *) -1/r /. FindRoot[{r^2 * QPochhammer[r^2] * QPochhammer[-1/(r*s), r^2] * QPochhammer[-s/r, r^2] / ((r + s)*(1 + r*s)) == 1, -1 + r/(r + s) + 1/(1 + r*s) + (QPolyGamma[0, Log[-1/(r*s)]/Log[r^2], r^2] - QPolyGamma[0, Log[-s/r]/Log[r^2], r^2])/Log[r^2] == 1}, {r, -1/4}, {s, 1/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, (A^m+A^-m+x*O(x^n))*x^(m^2))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1-x^(2*m))*(1+A*x^(2*m-1))*(1+A^-1*x^(2*m-1)+x*O(x^n)))); polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 20 2011
STATUS
approved