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
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.
Crossrefs
Cf. A158690.
Programs
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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 *)
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
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