cp's OEIS Frontend

A317362 Expansion of e.g.f. exp(exp(x/(1 + x)) - 1).

%I A317362 #7 Jul 26 2018 16:27:03
%S A317362 1,1,0,-1,3,-8,23,-89,556,-4773,44425,-397670,3060577,-12448655,
%T A317362 -235761640,9571505555,-241952653453,5424619822460,-116900288145113,
%U A317362 2494797839905055,-53406941947725348,1152770311462756071,-25109138533156554399,550613923917090815374,-12088287036694435407999
%N A317362 Expansion of e.g.f. exp(exp(x/(1 + x)) - 1).
%C A317362 Inverse Lah transform of the Bell numbers (A000110).
%H A317362 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A317362 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*Bell(k)*n!/k!, where Bell() = A000110.
%p A317362 a:= proc(n) option remember; add((-1)^(n-k)*n!/k!*
%p A317362       binomial(n-1, k-1)*combinat[bell](k), k=0..n)
%p A317362     end:
%p A317362 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 26 2018
%t A317362 nmax = 24; CoefficientList[Series[Exp[Exp[x/(1 + x)] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A317362 Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] BellB[k] n!/k!, {k, 0, n}], {n, 0, 24}]
%Y A317362 Cf. A000110, A084357.
%K A317362 sign
%O A317362 0,5
%A A317362 _Ilya Gutkovskiy_, Jul 26 2018