A370516 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+3,i+1)*(-1)^i for n >= 0, 0 <= k <= n.

1, -5, 1, 7, -4, 1, -3, 3, -3, 1, 0, 0, 0, -2, 1, 0, 0, 0, -2, -1, 1, 0, 0, 0, -2, -3, 0, 1, 0, 0, 0, -2, -5, -3, 1, 1, 0, 0, 0, -2, -7, -8, -2, 2, 1, 0, 0, 0, -2, -9, -15, -10, 0, 3, 1, 0, 0, 0, -2, -11, -24, -25, -10, 3, 4

Offset: 0

Author

Igor Victorovich Statsenko, Feb 21 2024

Keywords

• tabl
• sign

Comments

Generalized binomial coefficients of the second order.

Examples

			n\k   0    1    2    3    4    5    6
0:    1
1:   -5    1
2:    7   -4    1
3:   -3    3   -3    1
4:    0    0    0   -2    1
5:    0    0    0   -2   -1    1
6:    0    0    0   -2   -3    0    1
		

Links

• Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Crossrefs

For m=0 the formula gives the sequence A007318; for m=1 the formula gives the sequence A159854. In this case, we assume that A007318 consists of generalized binomial coefficients of order zero and A159854 consists of generalized binomial coefficients of order one.

Programs

Maple

C:=(n,k)->n!/(k!*(n-k)!) : T:=(m,n,k)->sum(C(n+1,n-k-r)*Stirling2(r+m+1,r+1)*((-1)^r), r=0..n-k) : m:=2 : seq(seq T(m,n,k), k=0..n), n=0..10);

Formula

T(n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^i where m = 2 for n >= 0, 0 <= k <= n.