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%I #8 Apr 18 2016 06:38:07
%S 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,
%T 265,689,694,355,100,15,1,0,1854,5053,5453,3094,1015,196,21,1,0,14833,
%U 42048,48082,29596,10899,2492,350,28,1
%N 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.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>
%e Triangle starts:
%e 1,
%e 0, 1,
%e 0, 0, 1,
%e 0, 1, 1, 1,
%e 0, 2, 5, 3, 1,
%e 0, 9, 20, 17, 6, 1,
%e 0, 44, 109, 100, 45, 10, 1,
%e 0, 265, 689, 694, 355, 100, 15, 1.
%p A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
%p seq(seq(A269954(n, k), k=0..n), n=0..9);
%t Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]]
%Y A000255 (row sums), A000166(col. 1), A000217 (diag. n,n-1), A133252 (diag. n,n-2).
%K nonn,tabl
%O 0,12
%A _Peter Luschny_, Apr 12 2016
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