A356654 - OEIS

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A356654

Triangle read by rows. T(n, k) = k! * Sum_{j=k..n} Lah(n, j) * Stirling2(j, k), where Lah(n, k) = A271703(n, k).

1, 0, 1, 0, 3, 2, 0, 13, 18, 6, 0, 73, 158, 108, 24, 0, 501, 1510, 1590, 720, 120, 0, 4051, 15962, 23040, 15960, 5400, 720, 0, 37633, 186270, 345786, 325920, 168000, 45360, 5040, 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

The same construction with Stirling1 in place of Stirling2 gives A225479, the ordered Stirling cycle numbers.

LINKS

Table of n, a(n) for n=0..44.

EXAMPLE

Triangle T(n, k) begins:

[0] 1;

[1] 0, 1;

[2] 0, 3, 2;

[3] 0, 13, 18, 6;

[4] 0, 73, 158, 108, 24;

[5] 0, 501, 1510, 1590, 720, 120;

[6] 0, 4051, 15962, 23040, 15960, 5400, 720;

[7] 0, 37633, 186270, 345786, 325920, 168000, 45360, 5040;

[8] 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320;

MAPLE

L := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1) * n! / k!):

T := (n, k) -> k! * add(L(n, j) * Stirling2(j, k), j = k..n):

seq(seq(T(n, k), k = 0..n), n = 0..9);

MATHEMATICA

T[n_, k_] := k! * Sum[Binomial[n, j] * FactorialPower[n - 1, n - j] * StirlingS2[j, k], {j, k, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 01 2022 *)

CROSSREFS

Cf. A271703, A048993, A225479, A000262 (column 1), A052838 (column 2), A084358 (row sums).

Sequence in context: A355360 A067346 A360282 * A282423 A111541 A371025

Adjacent sequences: A356651 A356652 A356653 * A356655 A356656 A356657

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Sep 01 2022

STATUS

approved