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A271700 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

%I A271700 #7 Apr 18 2016 06:38:22
%S A271700 1,1,1,1,2,3,1,3,6,16,1,4,10,30,115,1,5,15,50,205,1021,1,6,21,77,336,
%T A271700 1750,10696,1,7,28,112,518,2814,17766,128472,1,8,36,156,762,4308,
%U A271700 28050,207942,1734447,1,9,45,210,1080,6342,42528,322860,2746815,25937683
%N A271700 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.
%e A271700 Triangle starts:
%e A271700 [1]
%e A271700 [1, 1]
%e A271700 [1, 2, 3]
%e A271700 [1, 3, 6,  16]
%e A271700 [1, 4, 10, 30,  115]
%e A271700 [1, 5, 15, 50,  205, 1021]
%e A271700 [1, 6, 21, 77,  336, 1750, 10696]
%e A271700 [1, 7, 28, 112, 518, 2814, 17766, 128472]
%p A271700 T := (n,k) -> add(abs(Stirling1(k,j))*binomial(-j-1,-n-1)*(-1)^(n-j),j=0..n);
%p A271700 seq(seq(T(n,k), k=0..n), n=0..9);
%t A271700 Flatten[Table[Sum[(-1)^(n-j)Binomial[-j-1,-n-1] Abs[StirlingS1[k,j]],{j,0,n}], {n,0,9},{k,0,n}]]
%Y A271700 A000027 (col. 1), A000217, A161680 (col. 2), A005581 (col. 3), A211210 (diag. n,n), A211211 (diag. n,n-1).
%K A271700 nonn,tabl
%O A271700 0,5
%A A271700 _Peter Luschny_, Apr 14 2016