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A269954 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.

%I A269954 #23 Feb 13 2025 07:54:17
%S A269954 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 A269954 265,689,694,355,100,15,1,0,1854,5053,5453,3094,1015,196,21,1,0,14833,
%U A269954 42048,48082,29596,10899,2492,350,28,1
%N A269954 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 A269954 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>
%e A269954 Triangle starts:
%e A269954   1;
%e A269954   0,   1;
%e A269954   0,   0,   1;
%e A269954   0,   1,   1,   1;
%e A269954   0,   2,   5,   3,   1;
%e A269954   0,   9,  20,  17,   6,   1;
%e A269954   0,  44, 109, 100,  45,  10,  1;
%e A269954   0, 265, 689, 694, 355, 100, 15, 1;
%p A269954 A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
%p A269954 seq(seq(A269954(n, k), k=0..n), n=0..9);
%t A269954 Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]]
%o A269954 (PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k)));
%o A269954 for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ _Seiichi Manyama_, Feb 13 2025
%Y A269954 A000255 (row sums), A000217 (diag. n,n-1), A133252 (diag. n,n-2).
%Y A269954 Columns k=0..4 give A000007, A000166(n-1), A300490(n-1), A381067(n-1), A381068(n-1).
%Y A269954 Cf. A269951, A269953.
%K A269954 nonn,tabl
%O A269954 0,12
%A A269954 _Peter Luschny_, Apr 12 2016