A355162 - OEIS

%I #10 Jun 27 2022 03:18:56

%S 1,6,52,568,7312,107360,1760576,31760256,623137024,13179872768,

%T 298391335936,7189153167360,183428957442048,4935794590572544,

%U 139571328018628608,4134634425826115584,127966201403431518208,4127825849826169716736,138477447400991610896384,4822002684952714247929856

%N a(n) = exp(-1) * Sum_{k>=0} (4*k + 2)^n / k!.

%F E.g.f.: exp(exp(4*x) + 2 x - 1).

%F a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).

%F a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n+k) * Bell(k).

%F a(n) = 2^n * A126390(n). - _Vaclav Kotesovec_, Jun 22 2022

%F a(n) ~ 4^n * n^(n + 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/2)). - _Vaclav Kotesovec_, Jun 27 2022

%t nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 2 x - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

%t Table[Sum[Binomial[n, k] 2^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]

%Y Cf. A000110, A005493, A126390, A284859, A284864, A285064, A355163.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jun 22 2022