A301833 G.f. A(x) satisfies: A(x) = 1/(1 - 2*x*A(x)/(1 - 2*x*A(x)/(1 - 4*x*A(x)/(1 - 4*x*A(x)/(1 - 6*x*A(x)/(1 - 6*x*A(x)/(1 - ...))))))), a continued fraction.

1, 2, 12, 104, 1104, 13472, 183488, 2749056, 44996864, 802443776, 15579089920, 329170937856, 7562372632576, 188526816632832, 5083702487990272, 147676990509580288, 4600624321049722880, 153012055369679241216, 5409813656756850262016, 202534832564335070937088, 8001606648308588124045312

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Index entries for sequences related to factorial numbers

Formula

a(n) = [x^n] (Sum_{k>=0} A000165(k)*x^k)^(n+1)/(n + 1).

a(n) ~ sqrt(Pi) * (2*n)^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Nov 05 2021

Example

G.f. A(x) = 1 + 2*x + 12*x^2 + 104*x^3 + 1104*x^4 + 13472*x^5 + 183488*x^6 + 2749056*x^7 + 44996864*x^8 + ...
log(A(x)) = 2*x + 20*x^2/2 + 248*x^3/3 + 3472*x^4/4 + 53152*x^5/5 + 878144*x^6/6 + ... + A293471(n)*x^n/n + ...

Mathematica

Table[SeriesCoefficient[(1 + Sum[(2*k)!!*x^k, {k, 1, n}])^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)

Crossrefs

Cf. A000165, A088368, A293471, A301363.

Sequence in context: A152254 A157328 A061632 * A320899 A194951 A104533

Adjacent sequences: A301830 A301831 A301832 * A301834 A301835 A301836

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 27 2018

Status

approved