A376582 Triangle of generalized Stirling numbers.

1, 5, 1, 26, 7, 1, 154, 47, 9, 1, 1044, 342, 74, 11, 1, 8028, 2754, 638, 107, 13, 1, 69264, 24552, 5944, 1066, 146, 15, 1, 663696, 241128, 60216, 11274, 1650, 191, 17, 1, 6999840, 2592720, 662640, 127860, 19524, 2414, 242, 19, 1, 80627040, 30334320, 7893840, 1557660, 245004, 31594, 3382, 299, 21, 1

Offset: 0

Author

Igor Victorovich Statsenko, Sep 29 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]       1;
[1]       5,       1;
[2]      26,       7,       1;
[3]     154,      47,       9,        1;
[4]    1044,     342,      74,       11,       1;
[5]    8028,    2754,     638,      107,      13,     1;
[6]   69264,   24552,    5944,     1066,     146,    15,    1;
[7]  663696,  241128,   60216,    11274,    1650,   191,   17,    1;
		

Crossrefs

Column k: A001705 (k=0), A001711 (k=1), A001716 (k=2), A001721 (k=3), A051524 (k=4), A051545 (k=5), A051560 (k=6).

Cf. A094587 and A173333 for m=0.

Programs

Maple

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k): m:=1: seq(seq(T(m,n,k), k=0..n), n=0..10);

Formula

T(m,n,k) = Sum_{i=0..n-k} Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!, for m=1.