A351423 -id:A351423 - cp's OEIS frontend

A351420 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 3, -1, 24, 1, -3, 8, -13, 8, 120, 1, -4, 16, -48, 77, -26, 720, 1, -5, 27, -124, 386, -576, 194, 5040, 1, -6, 41, -259, 1270, -3905, 5219, -1142, 40320, 1, -7, 58, -471, 3244, -16243, 47701, -55567, 9736, 362880

Offset: 1

Seiichi Manyama, Feb 11 2022

			Square array begins:
    1,   1,    1,     1,      1,      1, ...
    1,   0,   -1,    -2,     -3,     -4, ...
    2,   1,    3,     8,     16,     27, ...
    6,  -1,  -13,   -48,   -124,   -259, ...
   24,   8,   77,   386,   1270,   3244, ...
  120, -26, -576, -3905, -16243, -50375, ...

Columns k=1..5 give A000142(n-1), (-1)^(n-1) * A089064(n), A351421, A351422, A351423.
Main diagonal gives A351424.
Cf. A111933, A351429.

T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Feb 11 2022 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));

T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.

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

Original entry on oeis.org

1, -1, -3, -11, -41, -75, 1540, 37725, 657715, 10551750, 163089430, 2407275470, 31865298262, 290682880132, -2479867505029, -267542605513289, -11438897571729494, -404343336811199242, -13192591498632627584, -410340915410006575406, -12233989907129223814578

Offset: 0

Seiichi Manyama, Feb 11 2022

sign

Column k=5 of A351429.
Cf. A000587, A130410, A351427.
Cf. A000357, A351423.

g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/((g@@5)(x)+1), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 11 2022

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 *)
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 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1)))

T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, 5);

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!.

cp's OEIS Frontend

A351420 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).

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

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