A355781 - OEIS

%I #22 Nov 16 2022 08:53:16

%S 1,2,14,166,2854,64854,1839622,62688406,2497159302,113932356630,

%T 5860555367814,335639363668118,21184456464757894,1461163816568091926,

%U 109351697864286862214,8825909581376322510230,764231343305480319046278,70670539764733828998689302

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

%H Alois P. Heinz, <a href="/A355781/b355781.txt">Table of n, a(n) for n = 0..347</a>

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

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

%F From _Vaclav Kotesovec_, Jul 18 2022: (Start)

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

%F 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)

%p b:= proc(n, m) option remember; `if`(n=0,

%p      2^m*(m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..21);  # _Alois P. Heinz_, Jul 29 2022

%t  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]];

%t a[n_] := b[n, 0];

%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Nov 16 2022, after _Alois P. Heinz_ *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(1-exp(x))))))

%o (PARI) a(n) = sum(k=0, n, 2^k*(k+1)^(k-1)*stirling(n, k, 2));

%Y Cf. A052880, A351276, A355788.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 16 2022