A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

1, 3, 1, 13, 7, 1, 73, 50, 12, 1, 501, 400, 125, 18, 1, 4051, 3609, 1335, 255, 25, 1, 37633, 36463, 15214, 3485, 460, 33, 1, 394353, 408694, 186949, 48769, 7805, 763, 42, 1, 4596553, 5036792, 2479602, 714364, 131299, 15708, 1190, 52, 1, 58941091, 67714809, 35419350, 11045558, 2256933, 312375, 29190, 1770, 63, 1, 824073141, 986271823, 543025851, 180766890, 40194965, 6221397, 676893, 50970, 2535, 75, 1

Offset: 0

Author

Igor Victorovich Statsenko, Oct 07 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]        1;
[1]        3,        1;
[2]       13,        7,        1;
[3]       73,       50,       12,       1;
[4]      501,      400,      125,      18,       1;
[5]     4051,     3609,     1335,     255,      25,       1;
[6]    37633,    36463,    15214,    3485,     460,      33,      1;
[7]   394353,   408694,   186949,   48769,    7805,     763,     42,    1;
[8]  4596553,  5036792,  2479602,  714364,  131299,   15708,   1190,   52,     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. 19-12. In Russian.

Crossrefs

A000262 (column 0), A052852 (row sums).

Triangle for m=0: A130534.

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:=1: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 = 1.