A370518 Triangle of numbers read by rows: T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(n,i) where TC(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, 14, -9, 1, -18, 29, -12, 1, 0, -22, 35, -14, 1, 0, -26, 15, 25, -15, 1, 0, -60, 4, 75, -5, -15, 1, 0, -204, -56, 259, 70, -56, -14, 1, 0, -912, -484, 1092, 609, -168, -126, -12, 1, 0, -5040, -3708, 5480, 4599, -231, -882, -210, -9, 1, 0, -33120, -30024, 31820, 36350, 3675, -6027, -2370, -300, -5, 1

Offset

0,2

Comments

Generalized Stirling numbers of the first kind of the second order.

Links

Table of n, a(n) for n=0..65.

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.

Formula

T(n,k) = Sum_{i=0..n} binomial(n,i)*(n-i)!*Stirling1(i,k)*TC(m,n,i) where TC(m,n,k) = Sum_{i=0..n-k} binomial(n+1,n-k-i)*Stirling2(i+m+1,i+1)*(-1)^(i),m = 2 for n >= 0.

Example

n\k     0     1     2     3     4     5     6
0:      1
1:     -5     1
2:     14    -9     1
3:    -18    29   -12     1
4:      0   -22    35   -14     1
5:      0   -26    15    25   -15     1
6:      0   -60     4    75    -5   -15     1

Maple

C:=(n, k)->n!/(k!*(n-k)!) : T0:=(m, n, k)->sum(C(n+1, n-k-p)*Stirling2(p+m+1, p+1)*((-1)^p), p=0..n-k) : T:=(m, n, k)->sum(C(n, r)*(n-r)!*Stirling1(r, k)*T0(m, n, r), r=0..n)  m:=2 : seq(seq T(m, n, k), k=0..n), n=0..10);

Crossrefs

For m=0 the formula gives the sequence A130534; for m=1 the formula gives the sequence A094645. In this case, we assume that A130534 consists of generalized Stirling numbers of the first kind of zero order, and A094645 consists of generalized Stirling numbers of the first kind of the first order.

Sequence in context: A067558 A104792 A120393 * A094368 A295574 A087727

Adjacent sequences: A370515 A370516 A370517 * A370519 A370520 A370521

Keyword

tabl,sign

Author

Igor Victorovich Statsenko, Feb 21 2024

Status

approved