A269954 - OEIS

%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