OFFSET
0,2
LINKS
Igor Victorovich Statsenko, Relationships of āPā-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.
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.
EXAMPLE
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;
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);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Igor Victorovich Statsenko, Oct 07 2024
STATUS
approved