A325873 - OEIS

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

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OFFSET

0,13

LINKS

Table of n, a(n) for n=0..65.

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)

EXAMPLE

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]

MATHEMATICA

p[n_] := Sum[Abs[StirlingS1[n, k]] FactorialPower[x, k], {k, 0, n}];

Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

PROG

(SageMath)

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)]

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

CROSSREFS

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

Cf. A079642 (variant), A129062, A325872.

Sequence in context: A372762 A019974 A344373 * A385343 A363973 A046781

Adjacent sequences: A325870 A325871 A325872 * A325874 A325875 A325876

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jun 27 2019

STATUS

approved