A302017 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^(2*k-1))).

1, 1, 2, 3, 6, 11, 21, 39, 73, 137, 257, 482, 903, 1693, 3173, 5948, 11149, 20899, 39174, 73430, 137641, 258002, 483614, 906513, 1699219, 3185111, 5970352, 11191163, 20977346, 39321116, 73705711, 138158128, 258971363, 485430483, 909918190, 1705601814, 3197075934, 5992778881, 11233201667

Offset: 0

Author

Ilya Gutkovskiy, Mar 30 2018

Keywords

• nonn

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Crossrefs

Antidiagonal sums of absolute values of A286352.

Cf. A000700, A299106, A299208.

Programs

Mathematica

nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[1/(1 - x QPochhammer[x^2]^2/(QPochhammer[x] QPochhammer[x^4])), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + (-x)^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A000700(k-1)*a(n-k).

a(n) ~ c / r^n, where r = 0.5334880525001986092393688937248506539793821912... is the root of the equation 1 + r - r^2 * QPochhammer(-1/r, r^2) = 0 and c = 0.48000092330632206397886602198643227268597451507794232644772186731542555975... = (2*(1 + r)*Log[r])/(2*(2 + r)*Log[r] + (1 + r)*Log[1 - r^2] + (1 + r) * QPolyGamma[Log[-1/r] / Log[r^2], r^2] + 4*r^4*Log[r] * Derivative[0,1][QPochhammer][-1/r, r^2]). - Vaclav Kotesovec, Mar 31 2018