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A028248 Expansion of exp(exp(exp(x) - 1 - x) - 1).

%I A028248 #23 Oct 21 2023 09:53:24
%S A028248 1,0,1,1,7,21,126,638,4264,28983,226018,1872300,16940661,163461455,
%T A028248 1688378030,18501971647,214749109038,2628228896227,33832314246857,
%U A028248 456730760934125,6451399211318995,95135434800384144,1461771954435844296,23360315241127222572
%N A028248 Expansion of exp(exp(exp(x) - 1 - x) - 1).
%C A028248 For p prime, a(p) == 1 (mod p) and a(p+1) == 1 (mod p). - _Mélika Tebni_, Mar 22 2022
%H A028248 Alois P. Heinz, <a href="/A028248/b028248.txt">Table of n, a(n) for n = 0..495</a>
%F A028248 Row sums of A352607. - _Mélika Tebni_, Mar 22 2022
%e A028248 From _Mélika Tebni_, Mar 22 2022: (Start)
%e A028248 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 A028248 h:= proc(n, m) option remember;
%p A028248      `if`(n=0, 1, h(n-1, m+1)+m*h(n-1, m))
%p A028248     end:
%p A028248 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A028248       a(n-j)*binomial(n-1, j-1)*h(j, -1), j=1..n))
%p A028248     end:
%p A028248 seq(a(n), n=0..23);  # _Alois P. Heinz_, Apr 14 2023
%t A028248 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 A028248 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(exp(x) - 1 - x) - 1))) \\ _Michel Marcus_, Mar 22 2022
%Y A028248 Cf. A000110, A137375, A352607.
%K A028248 nonn
%O A028248 0,5
%A A028248 _N. J. A. Sloane_