A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.

1, 7, 65, 743, 9921, 150151, 2526593, 46615783, 933072513, 20093861895, 462440842177, 11310514854375, 292627518129985, 7976748158144647, 228308400790500097, 6840702405678586343, 214000748166439723265, 6973447420429351808007, 236204029044752265931585, 8300724166287243795922151

Offset

0,2

Links

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

Formula

E.g.f.: exp(exp(4*x) + 3 x - 1).

a(0) = 1; a(n) = 3 * 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) * 3^(n-k) * 4^k * Bell(k).

a(n) ~ Bell(n) * (4 + 3*LambertW(n)/n)^n. - Vaclav Kotesovec, Jun 22 2022

a(n) ~ 4^n * n^(n + 3/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 3/4)). - Vaclav Kotesovec, Jun 27 2022

Mathematica

nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 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] 3^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]

Crossrefs

Cf. A000110, A005494, A126390, A284859, A284864, A285064, A355162.

Sequence in context: A099342 A051550 A371393 * A300488 A085283 A069015

Adjacent sequences: A355160 A355161 A355162 * A355164 A355165 A355166

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 22 2022

Status

approved