A355162 a(n) = exp(-1) * Sum_{k>=0} (4*k + 2)^n / k!.

1, 6, 52, 568, 7312, 107360, 1760576, 31760256, 623137024, 13179872768, 298391335936, 7189153167360, 183428957442048, 4935794590572544, 139571328018628608, 4134634425826115584, 127966201403431518208, 4127825849826169716736, 138477447400991610896384, 4822002684952714247929856

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

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

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

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

a(n) = 2^n * A126390(n). - Vaclav Kotesovec, Jun 22 2022

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

Mathematica

nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 2 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
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}]
Table[Sum[Binomial[n, k] 2^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]

Crossrefs

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

Sequence in context: A365194 A365755 A232821 * A127133 A243249 A075756

Adjacent sequences: A355159 A355160 A355161 * A355163 A355164 A355165

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 22 2022

Status

approved