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
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.
Programs
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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.