A237653 -id:A237653 - cp's OEIS frontend

A343579 a(n) = Sum_{k=0..floor(n/2)} |Stirling1(n - k, k)|.

1, 0, 1, 1, 3, 9, 36, 176, 1030, 7039, 55098, 486346, 4780445, 51787405, 613045468, 7873065045, 109021348618, 1619197654575, 25675094145535, 432908683794379, 7733991639921585, 145933532935469016, 2900112108790279902, 60543749629794205640, 1324677739541613767983

Offset: 0

Peter Luschny, Apr 20 2021

nonn

Equals antidiagonal sums of the triangle of unsigned Stirling numbers of the first kind (A132393).

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A132393, A320963.

Variant: A237653.

Table[Sum[Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 09 2022 *)

a(n) = sum(k=0, n\2, abs(stirling(n-k, k, 1))); \\ Michel Marcus, Apr 22 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+x))) \\ Seiichi Manyama, Apr 08 2022

a(n) ~ n! / n^2. - Vaclav Kotesovec, Apr 09 2022

A331327 T(n, k) = [x^k] Pochhammer(x, n-k) for n >= 0, 0 <= k <= floor(n/2). Irregular triangle read by rows.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 6, 3, 0, 24, 11, 1, 0, 120, 50, 6, 0, 720, 274, 35, 1, 0, 5040, 1764, 225, 10, 0, 40320, 13068, 1624, 85, 1, 0, 362880, 109584, 13132, 735, 15, 0, 3628800, 1026576, 118124, 6769, 175, 1, 0, 39916800, 10628640, 1172700, 67284, 1960, 21

Offset: 0

Peter Luschny, Jan 25 2020

			Triangle starts:
[ 0] 1;
[ 1] 0;
[ 2] 0, 1;
[ 3] 0, 1;
[ 4] 0, 2,     1;
[ 5] 0, 6,     3;
[ 6] 0, 24,    11,    1;
[ 7] 0, 120,   50,    6;
[ 8] 0, 720,   274,   35,   1;
[ 9] 0, 5040,  1764,  225,  10;
[10] 0, 40320, 13068, 1624, 85, 1;

Row sums are: 1, 0, A237653.

Cf. A132393.

A331327row := n -> seq(coeff(expand(pochhammer(x, n-k)), x, k), k=0..n/2):
seq(A331327row(n), n=0..13);

T[n_, k_] := Abs[StirlingS1[n - k, k]];
Table[T[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

Rows are the antidiagonals of A132393.

cp's OEIS Frontend

A343579 a(n) = Sum_{k=0..floor(n/2)} |Stirling1(n - k, k)|.

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