A069138 -id:A069138

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A360174

Triangle read by rows. T(n, k) = (k + 1) * abs(Stirling1(n, k)).

+10
2

1, 0, 2, 0, 2, 3, 0, 4, 9, 4, 0, 12, 33, 24, 5, 0, 48, 150, 140, 50, 6, 0, 240, 822, 900, 425, 90, 7, 0, 1440, 5292, 6496, 3675, 1050, 147, 8, 0, 10080, 39204, 52528, 33845, 11760, 2254, 224, 9, 0, 80640, 328752, 472496, 336420, 134694, 31752, 4368, 324, 10

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

Triangle T(n, k) starts:

[0] 1;

[1] 0, 2;

[2] 0, 2, 3;

[3] 0, 4, 9, 4;

[4] 0, 12, 33, 24, 5;

[5] 0, 48, 150, 140, 50, 6;

[6] 0, 240, 822, 900, 425, 90, 7;

[7] 0, 1440, 5292, 6496, 3675, 1050, 147, 8;

[8] 0, 10080, 39204, 52528, 33845, 11760, 2254, 224, 9;

MAPLE

T := (n, k) -> (k + 1)*abs(Stirling1(n, k)):

for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

CROSSREFS

Cf. A208529 (column 1), A006002 (subdiagonal), A000774 (row sums).

Cf. A069138 (Stirling2 counterpart), A360205 (Lah counterpart).

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Feb 08 2023

STATUS

approved

A360205

Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).

+10
2

1, 0, 2, 0, 4, 3, 0, 12, 18, 4, 0, 48, 108, 48, 5, 0, 240, 720, 480, 100, 6, 0, 1440, 5400, 4800, 1500, 180, 7, 0, 10080, 45360, 50400, 21000, 3780, 294, 8, 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9, 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10

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

OFFSET

0,3

COMMENTS

A refinement of the number of partial permutations of an n-set (A002720).

Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

Triangle T(n, k) starts:

[0] 1;

[1] 0, 2;

[2] 0, 4, 3;

[3] 0, 12, 18, 4;

[4] 0, 48, 108, 48, 5;

[5] 0, 240, 720, 480, 100, 6;

[6] 0, 1440, 5400, 4800, 1500, 180, 7;

[7] 0, 10080, 45360, 50400, 21000, 3780, 294, 8;

[8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;

MAPLE

T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):

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