A028248 - OEIS

%I #23 Oct 21 2023 09:53:24

%S 1,0,1,1,7,21,126,638,4264,28983,226018,1872300,16940661,163461455,

%T 1688378030,18501971647,214749109038,2628228896227,33832314246857,

%U 456730760934125,6451399211318995,95135434800384144,1461771954435844296,23360315241127222572

%N Expansion of exp(exp(exp(x) - 1 - x) - 1).

%C For p prime, a(p) == 1 (mod p) and a(p+1) == 1 (mod p). - _Mélika Tebni_, Mar 22 2022

%H Alois P. Heinz, <a href="/A028248/b028248.txt">Table of n, a(n) for n = 0..495</a>

%F Row sums of A352607. - _Mélika Tebni_, Mar 22 2022

%e From _Mélika Tebni_, Mar 22 2022: (Start)

%e a(11) = Sum_{k=0..5} (-1)^k*Bell(k)*A137375(11, k) = 1*(0) - 1*(-1) + 2*(1012) - 5*(-22935) + 15*(56980) - 52*(-17325) = 1872300. (End)

%p h:= proc(n, m) option remember;

%p      `if`(n=0, 1, h(n-1, m+1)+m*h(n-1, m))

%p     end:

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       a(n-j)*binomial(n-1, j-1)*h(j, -1), j=1..n))

%p     end:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Apr 14 2023

%t A352607[n_, k_] := BellB[k]*Sum[(-1)^(k + j)*Binomial[n, n - k + j]* StirlingS2[n - k + j, j], {j, 0, k}]; a[n_] := Sum[A352607[n, k], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Oct 21 2023 *)

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(exp(x) - 1 - x) - 1))) \\ _Michel Marcus_, Mar 22 2022

%Y Cf. A000110, A137375, A352607.

%K nonn

%O 0,5

%A _N. J. A. Sloane_