A269954 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.
1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 5, 3, 1, 0, 9, 20, 17, 6, 1, 0, 44, 109, 100, 45, 10, 1, 0, 265, 689, 694, 355, 100, 15, 1, 0, 1854, 5053, 5453, 3094, 1015, 196, 21, 1, 0, 14833, 42048, 48082, 29596, 10899, 2492, 350, 28, 1
Offset: 0
Examples
Triangle starts: 1; 0, 1; 0, 0, 1; 0, 1, 1, 1; 0, 2, 5, 3, 1; 0, 9, 20, 17, 6, 1; 0, 44, 109, 100, 45, 10, 1; 0, 265, 689, 694, 355, 100, 15, 1;
Links
- Peter Luschny, Extensions of the binomial
Crossrefs
Programs
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Maple
A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n): seq(seq(A269954(n, k), k=0..n), n=0..9);
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Mathematica
Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]] -
PARI
T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k))); for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ Seiichi Manyama, Feb 13 2025