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Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).
6

%I #21 May 13 2022 18:21:31

%S 1,0,1,3,12,70,465,3591,31948,319068,3539385,43205635,575312826,

%T 8298867798,128921967265,2145837600375,38097353658120,718657756980376,

%U 14354000800751313,302625047150614179,6716038666999745710,156498725047355717250,3820426102008414736761

%N Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).

%H Seiichi Manyama, <a href="/A346888/b346888.txt">Table of n, a(n) for n = 0..442</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,2) * a(n-k).

%F a(n) ~ n! / ((1 + LambertW(1/sqrt(2))) * 2^(n+1) * LambertW(1/sqrt(2))^n). - _Vaclav Kotesovec_, Aug 08 2021

%F a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k)/(2^k * (n-2*k)!). - _Seiichi Manyama_, May 13 2022

%t nmax = 22; CoefficientList[Series[1/(1 - x^2 Exp[x]/2), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^2*exp(x)/2))) \\ _Michel Marcus_, Aug 06 2021

%o (PARI) a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(2^k*(n-2*k)!)); \\ _Seiichi Manyama_, May 13 2022

%Y Column k=2 of A351703.

%Y Cf. A006153, A346889, A346890, A346893.

%Y Cf. A000217, A133189, A308946, A353998.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Aug 06 2021