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a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).
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%I #7 Jun 20 2022 03:14:34

%S 1,3,19,181,2299,36501,695427,15457709,392672651,11221959685,

%T 356339728243,12446649786429,474273933636411,19577992095770837,

%U 870345573347448803,41455153171478627533,2106173029315813515883,113694251997087087941925,6498401704686168598548435,392062852538564346207533789

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

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

%F a(n) ~ n! / (2*(1 + LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - _Vaclav Kotesovec_, Jun 20 2022

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

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

%Y Cf. A000670, A005493, A006155, A032032, A343673, A343674.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 25 2021