A209832 Expansion of the q-series Sum_{n>=0} (-1)^n*q^(n+1)*Product_{k = 1..n} (1 - q^(2*k-1)), q = exp(t), as a formal Taylor series in t.

1, 2, 12, 200, 6576, 353312, 28032192, 3077502080, 446470392576, 82695752049152, 19038594625539072, 5332477132779407360, 1785375992372231909376, 704147423230177089953792, 323094378183013059349757952, 170643791820813252598723543040

Offset

0,2

Comments

Compare with A158690.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Formula

E.g.f.: Sum_{n>=0} exp((n+1)*t) * Product_{k = 1..n} (exp((2*k-1)*t) - 1) = exp(t) + exp(2*t)*(exp(t) - 1) + exp(3*t)*(exp(t) - 1)*(exp(3*t) - 1) + ... = 1 + 2*t + 12*t^2/2! + 200*t^3/3! + ...

Conjectural S-fraction expansion for the o.g.f.:

1/(1-2*x/(1-4*x/(1-16*x/(1-20*x/(1-...-2*n(3*n-2)*x/(1-2*n(3*n-1)*x/(1-...

a(n) ~ 2^(3*n + 2) * 3^(n + 1/2) * n^(2*n + 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Oct 09 2023

Mathematica

nmax = 20; CoefficientList[Series[Sum[(-1)^n*Exp[x*(n + 1)] * Product[ (1 - Exp[(2*k - 1)*x]), {k, 1, n}], {n, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 09 2023 *)

Crossrefs

Cf. A158690.

Sequence in context: A367052 A245358 A317350 * A094157 A306715 A012598

Adjacent sequences: A209829 A209830 A209831 * A209833 A209834 A209835

Keyword

nonn,easy

Author

Peter Bala, Mar 14 2012

Status

approved