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A351428 Expansion of e.g.f. 1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1).

%I A351428 #17 Feb 09 2025 13:00:42
%S A351428 1,-1,-3,-11,-41,-75,1540,37725,657715,10551750,163089430,2407275470,
%T A351428 31865298262,290682880132,-2479867505029,-267542605513289,
%U A351428 -11438897571729494,-404343336811199242,-13192591498632627584,-410340915410006575406,-12233989907129223814578
%N A351428 Expansion of e.g.f. 1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1).
%F A351428 a(n) = T(n,5), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.
%p A351428 g:= x-> exp(x)-1:
%p A351428 a:= n-> n! * coeff(series(1/((g@@5)(x)+1), x, n+1), x, n):
%p A351428 seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 11 2022
%t A351428 T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, 5]; Array[a, 20, 0] (* _Amiram Eldar_, Feb 11 2022 *)
%t A351428 With[{nn=20},CoefficientList[Series[1/Exp[Exp[Exp[Exp[Exp[x]-1]-1]-1]-1],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 09 2025 *)
%o A351428 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1)))
%o A351428 (PARI) T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
%o A351428 a(n) = T(n, 5);
%Y A351428 Column k=5 of A351429.
%Y A351428 Cf. A000587, A130410, A351427.
%Y A351428 Cf. A000357, A351423.
%K A351428 sign
%O A351428 0,3
%A A351428 _Seiichi Manyama_, Feb 11 2022