A301833 - OEIS

%I #7 Nov 05 2021 05:41:08

%S 1,2,12,104,1104,13472,183488,2749056,44996864,802443776,15579089920,

%T 329170937856,7562372632576,188526816632832,5083702487990272,

%U 147676990509580288,4600624321049722880,153012055369679241216,5409813656756850262016,202534832564335070937088,8001606648308588124045312

%N 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.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

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

%F a(n) ~ sqrt(Pi) * (2*n)^(n + 1/2) / exp(n-1). - _Vaclav Kotesovec_, Nov 05 2021

%e 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 + ...

%e 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 + ...

%t 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 *)

%Y Cf. A000165, A088368, A293471, A301363.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 27 2018