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A370518
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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.
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1
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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
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OFFSET
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0,2
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COMMENTS
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Generalized Stirling numbers of the first kind of the second order.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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MAPLE
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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);
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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