A325872 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, -2, 1, 0, 7, -6, 1, 0, -35, 40, -12, 1, 0, 228, -315, 130, -20, 1, 0, -1834, 2908, -1485, 320, -30, 1, 0, 17582, -30989, 18508, -5005, 665, -42, 1, 0, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 0

Author

Peter Luschny, Jun 27 2019

Keywords

• sign
• tabl

Examples

			Triangle starts:
[0] [1]
[1] [0,       1]
[2] [0,      -2,        1]
[3] [0,       7,       -6,       1]
[4] [0,     -35,       40,     -12,        1]
[5] [0,     228,     -315,     130,      -20,      1]
[6] [0,   -1834,     2908,   -1485,      320,    -30,      1]
[7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1]
[8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1]
[9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1]
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
• Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 8.
• Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Crossrefs

Columns k=0..3 give A000007, (-1)^(n+1) * A003713(n), (-1)^n * A341587(n), (-1)^(n+1) * A341588(n).

Cf. A039814 (variant), A129062, A325873.

Programs

Mathematica

p[n_] := Sum[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, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025

Sage

def a_row(n):
    s = sum((-1)^(n-k)*stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)]

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)