A389008 Triangle read by rows: T(0,0) = 1, T(n,0) = 0 for n > 0; T(n,k) = (1/(k-1)!) * Sum_{j=k..n} binomial(n,j) * Stirling1(j,k) * (n-j+k-1)!, 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 14, 23, 6, 1, 0, 94, 120, 65, 10, 1, 0, 444, 1024, 525, 145, 15, 1, 0, 3828, 7532, 5719, 1645, 280, 21, 1, 0, 25584, 76236, 57764, 22449, 4200, 490, 28, 1, 0, 270576, 720360, 707048, 292320, 70329, 9324, 798, 36, 1

Offset

0,8

Links

Table of n, a(n) for n=0..54.

Formula

E.g.f. of column k: (1/k!) * ( log(1 + x) / (1 - x) )^k.

Sum_{k=0..n} (-1)^k * T(n,k) = A347978(n).

Example

Triangle begins:
  1;
  0,    1;
  0,    1,    1;
  0,    5,    3,    1;
  0,   14,   23,    6,    1;
  0,   94,  120,   65,   10,   1;
  0,  444, 1024,  525,  145,  15,  1;
  0, 3828, 7532, 5719, 1645, 280, 21, 1;
  ...

Mathematica

T[0, 0] := 1; T[n_, 0] := 0; T[n_, k_] := (1/(k - 1)!) Sum[Binomial[n, j] StirlingS1[j, k] (n - j + k - 1)!, {j, k, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Prog

(PARI) row(n) = if (n==0, [1], vector(n+1, k, k--; if (k==0, 0, sum(j=k, n, binomial(n, j) * stirling(j, k, 1) * (n-j+k-1)!)/(k-1)!))); \\ Michel Marcus, Sep 23 2025

Crossrefs

Row sums are A073478.

Columns are A000007, A024167, A389001, A389002, A389003.

Diagonals include: A000012, A000217, A241765.

Cf. A307419, A347978.

Sequence in context: A126853 A286127 A201654 * A265606 A368602 A132199

Adjacent sequences: A389005 A389006 A389007 * A389009 A389010 A389011

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Sep 22 2025

Status

approved