A325873 T(n, k) = [x^k] Sum_{k=0..n} |Stirling1(n, k)|*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 0, 8, 5, 10, 0, 1, 0, 26, 58, 15, 20, 0, 1, 0, 194, 217, 238, 35, 35, 0, 1, 0, 1142, 2035, 1008, 728, 70, 56, 0, 1, 0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1, 0, 81384, 134164, 85410, 47815, 9660, 4116, 210, 120, 0, 1

Offset: 0

Author

Peter Luschny, Jun 27 2019

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] [1]
[1] [0,    1]
[2] [0,    0,     1]
[3] [0,    1,     0,     1]
[4] [0,    1,     4,     0,    1]
[5] [0,    8,     5,    10,    0,    1]
[6] [0,   26,    58,    15,   20,    0,   1]
[7] [0,  194,   217,   238,   35,   35,   0,  1]
[8] [0, 1142,  2035,  1008,  728,   70,  56,  0, 1]
[9] [0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1]
		

Crossrefs

Columns k=0..2 give A000007, A089064, A341575.

Cf. A079642 (variant), A129062, A325872.

Programs

Mathematica

p[n_] := Sum[Abs[StirlingS1[n, k]] FactorialPower[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

PARI

T(n, k) = sum(j=k, n, abs(stirling(n, j, 1))*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025

Sage

def a_row(n):
    s = sum(stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..10)]

Formula

From Seiichi Manyama, Apr 18 2025: (Start)

T(n,k) = Sum_{j=k..n} |Stirling1(n,j)| * Stirling1(j,k).

E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 - log(1 - x)). (End)