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A271699 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

%I A271699 #7 Apr 18 2016 06:38:14
%S A271699 1,0,1,0,1,2,0,1,3,9,0,1,4,14,58,0,1,5,20,90,475,0,1,6,27,131,729,
%T A271699 4666,0,1,7,35,182,1064,7070,53116,0,1,8,44,244,1494,10284,79470,
%U A271699 684762,0,1,9,54,318,2034,14478,114918,1012368,9833391
%N A271699 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.
%e A271699 Triangle starts:
%e A271699 1,
%e A271699 0, 1,
%e A271699 0, 1, 2,
%e A271699 0, 1, 3, 9,
%e A271699 0, 1, 4, 14, 58,
%e A271699 0, 1, 5, 20, 90,  475,
%e A271699 0, 1, 6, 27, 131, 729,  4666,
%e A271699 0, 1, 7, 35, 182, 1064, 7070, 53116
%p A271699 T := (n,k) -> add(abs(Stirling1(k,j))*binomial(-j,-n)*(-1)^(n-j), j=0..n):
%p A271699 seq(seq(T(n,k), k=0..n), n=0..9);
%t A271699 Flatten[Table[Sum[(-1)^(n-j)Binomial[-j,-n] Abs[StirlingS1[k,j]],{j,0,n}], {n,0,9},{k,0,n}]]
%Y A271699 A000027 (col. 2), A000096 (col. 3), A247329 (diag. n,n).
%K A271699 nonn,tabl
%O A271699 0,6
%A A271699 _Peter Luschny_, Apr 14 2016