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A271702 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S2(k,j), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

%I A271702 #12 Sep 03 2019 08:36:27
%S A271702 1,1,1,1,2,3,1,3,6,13,1,4,10,26,71,1,5,15,45,140,456,1,6,21,71,246,
%T A271702 887,3337,1,7,28,105,399,1568,6405,27203,1,8,36,148,610,2584,11334,
%U A271702 51564,243203,1,9,45,201,891,4035,18849,91101,455712,2357356
%N A271702 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S2(k,j), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.
%F A271702 T(n,k) = Sum_{j=0..k} C(n,j) * S2(k,j). - _Alois P. Heinz_, Sep 03 2019
%e A271702 Triangle starts:
%e A271702 [1]
%e A271702 [1, 1]
%e A271702 [1, 2, 3]
%e A271702 [1, 3, 6, 13]
%e A271702 [1, 4, 10, 26, 71]
%e A271702 [1, 5, 15, 45, 140, 456]
%e A271702 [1, 6, 21, 71, 246, 887, 3337]
%e A271702 [1, 7, 28, 105, 399, 1568, 6405, 27203]
%p A271702 T := (n,k) -> add(Stirling2(k,j)*binomial(-j-1,-n-1)*(-1)^(n-j),j=0..n):
%p A271702 seq(seq(T(n,k), k=0..n), n=0..9);
%t A271702 Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1,-n-1] StirlingS2[k,j], {j,0,n}], {n,0,9}, {k,0,n}]]
%Y A271702 A000012 (col. 0), A000027 (col. 1), A000217 (col. 2), A008778 (col. 3), A122455 (diag. n,n), A134094 (diag. n,n-1).
%Y A271702 Cf. A048993.
%K A271702 nonn,tabl
%O A271702 0,5
%A A271702 _Peter Luschny_, Apr 14 2016