A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

1, 5, 1, 32, 11, 1, 248, 113, 18, 1, 2248, 1230, 263, 26, 1, 23272, 14534, 3765, 505, 35, 1, 270400, 186992, 55654, 9115, 865, 45, 1, 3479744, 2612000, 865186, 163779, 19110, 1372, 56, 1, 49079936, 39434448, 14235388, 3013164, 408569, 36288, 2058, 68, 1

Offset: 0

Author

Igor Victorovich Statsenko, Oct 14 2024

Keywords

• nonn

Comments

These numbers are a subset of the generalized Stirling numbers introduced in A370518. Therefore, we assume them to be numbers of the lower level of hierarchy with respect to A370518.

Examples

			[0]           1;
[1]           5,          1;
[2]          32,         11,         1;
[3]         248,        113,        18,        1;
[4]        2248,       1230,       263,       26,       1;
[5]       23272,      14534,      3765,      505,      35,      1;
[6]      270400,     186992,     55654,     9115,     865,     45,     1;
[7]     3479744,    2612000,    865186,   163779,   19110,   1372,    56,    1;
[8]    49079936,   39434448,  14235388,  3013164,  408569,  36288,  2058,   68,    1;
		

Links

• Igor Victorovich Statsenko, Relationships of "P"-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

Crossrefs

A361649 (row sums).

Triangle for m=0: A130534.

Triangle for m=1: A376863.

Programs

Maple

T := (m,n,k) -> add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n), i=0..n): m:=2: seq(seq(T(m,n,k), k=0..n), n=0..10);

Formula

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k)*binomial(n+m, i)*binomial(n, j)* binomial(j, i)*i!*m^(j-i), for m = 2.