A355781 E.g.f. satisfies log(A(x)) = 2 * (exp(x) - 1) * A(x).

1, 2, 14, 166, 2854, 64854, 1839622, 62688406, 2497159302, 113932356630, 5860555367814, 335639363668118, 21184456464757894, 1461163816568091926, 109351697864286862214, 8825909581376322510230, 764231343305480319046278, 70670539764733828998689302

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..347

Formula

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

a(n) = Sum_{k=0..n} 2^k * (k+1)^(k-1) * Stirling2(n,k).

From Vaclav Kotesovec, Jul 18 2022: (Start)

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

a(n) ~ sqrt(2*exp(1) + 1) * sqrt(log(1 + exp(-1)/2)) * n^(n-1) / (exp(n-1) * (log(exp(1) + 1/2) - 1)^n). (End)

Maple

b:= proc(n, m) option remember; `if`(n=0,
     2^m*(m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 29 2022

Mathematica

 b[n_, m_] := b[n, m] = If[n == 0, 2^m*(m + 1)^(m - 1), m*b[n - 1, m] + b[n - 1, m + 1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(1-exp(x))))))
(PARI) a(n) = sum(k=0, n, 2^k*(k+1)^(k-1)*stirling(n, k, 2));

Crossrefs

Cf. A052880, A351276, A355788.

Sequence in context: A245896 A338632 A124215 * A003582 A372796 A277373

Adjacent sequences: A355778 A355779 A355780 * A355782 A355783 A355784

Keyword

nonn

Author

Seiichi Manyama, Jul 16 2022

Status

approved