A336950 - OEIS

%I #15 Feb 20 2022 06:42:12

%S 1,1,6,42,392,4600,64752,1063216,19952256,421227648,9880951040,

%T 254960721664,7176891675648,218857588139008,7187394935347200,

%U 252897556424140800,9491754142468702208,378509920569294684160,15982018774576565649408,712306819507400060502016

%N E.g.f.: 1 / (1 - x * exp(2*x)).

%H Seiichi Manyama, <a href="/A336950/b336950.txt">Table of n, a(n) for n = 0..394</a>

%F a(n) = n! * Sum_{k=0..n} (2 * (n-k))^k / k!.

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

%F a(n) ~ n! * (2/LambertW(2))^n / (1 + LambertW(2)). - _Vaclav Kotesovec_, Aug 09 2021

%t nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!

%t Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

%o (PARI) seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ _Andrew Howroyd_, Aug 08 2020

%Y Column k=2 of A351790.

%Y Cf. A001787, A006153, A122704, A216794, A235328, A336947, A336951, A336952.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 08 2020