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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
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Generalized Stirling numbers of the first kind of the second order.
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.
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
KEYWORD
tabl,sign
AUTHOR
STATUS
approved

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Last modified July 23 04:43 EDT 2024. Contains 374544 sequences. (Running on oeis4.)