A111933 -id:A111933 - 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)!.

A351429 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + f^k(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, 1, -1, 1, -1, 2, 1, -1, 1, -6, 1, -1, 0, -1, 24, 1, -1, -1, 1, 1, -120, 1, -1, -2, 0, 1, -1, 720, 1, -1, -3, -4, 6, -2, 1, -5040, 1, -1, -4, -11, -2, 32, -9, -1, 40320, 1, -1, -5, -21, -41, 76, 115, -9, 1, -362880, 1, -1, -6, -34, -129, -75, 953, 172, 50, -1, 3628800

Offset: 0

Seiichi Manyama, Feb 11 2022

			Square array begins:
     1,  1,  1,   1,   1,    1,     1, ...
    -1, -1, -1,  -1,  -1,   -1,    -1, ...
     2,  1,  0,  -1,  -2,   -3,    -4, ...
    -6, -1,  1,   0,  -4,  -11,   -21, ...
    24,  1,  1,   6,  -2,  -41,  -129, ...
  -120, -1, -2,  32,  76,  -75,  -806, ...
   720,  1, -9, 115, 953, 1540, -3334, ...

Columns k=0..5 give A133942, A033999, A000587, A130410, A351427, A351428.
Main diagonal gives A351433.
Cf. A111933, A351420.

A:= (n, k)-> n!*(g->coeff(series(1/(1+(g@@k)(x)), x, n+1), x, n))(x->exp(x)-1):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # 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}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 11 2022 *)

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

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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