OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms
FORMULA
G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).
G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.9200517419026569743994130834319365190407162724411912701937027582419975778... is the root of the equation EllipticTheta(4, 0, 1/d) * d = 1 and c = 0.372842695601022868809531452599286285949969156503576039087883242107... = 2*Log[r]*QPochhammer[r] / (2*QPochhammer[r] * (Log[1 - r] + Log[r] + QPolyGamma[1, r]) + r*Log[r] * (r * Derivative[0, 1][QPochhammer][-1, r] - 2*Derivative[0, 1][QPochhammer][r, r])), where r = 1/d. Equivalently, c = EllipticTheta[4, 0, r]^2 / (r *(EllipticTheta[4, 0, r] - r * Derivative[0, 0, 1][EllipticTheta][4, 0, r])). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018
MAPLE
S:= series(1/(1-x/JacobiTheta4(0, x)), x, 51):
seq(coeff(S, x, n), n=0..50); # Robert Israel, Feb 02 2018
MATHEMATICA
nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 02 2018
STATUS
approved