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A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

%I A269952 #9 Apr 18 2016 06:37:45
%S A269952 1,0,1,0,2,1,0,4,5,1,0,8,19,9,1,0,16,65,55,14,1,0,32,211,285,125,20,1,
%T A269952 0,64,665,1351,910,245,27,1,0,128,2059,6069,5901,2380,434,35,1,0,256,
%U A269952 6305,26335,35574,20181,5418,714,44,1
%N A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.
%H A269952 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>
%F A269952 T(n, k) = S2(n+1, k+1) - S2(n, k+1).
%e A269952 1,
%e A269952 0, 1,
%e A269952 0, 2, 1,
%e A269952 0, 4, 5, 1,
%e A269952 0, 8, 19, 9, 1,
%e A269952 0, 16, 65, 55, 14, 1,
%e A269952 0, 32, 211, 285, 125, 20, 1,
%e A269952 0, 64, 665, 1351, 910, 245, 27, 1.
%p A269952 A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
%p A269952 seq(seq(A269952(n,k), k=0..n), n=0..9);
%t A269952 Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]
%Y A269952 Variant: A143494 (the main entry for this triangle).
%Y A269952 A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
%Y A269952 A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
%Y A269952 A049444, A136124, A143491 (matrix inverse).
%Y A269952 Cf. A048993, A269951.
%K A269952 nonn,tabl
%O A269952 0,5
%A A269952 _Peter Luschny_, Apr 10 2016