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
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]
Programs
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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
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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)