%I A131689
%S A131689 1,0,1,0,1,2,0,1,6,6,0,1,14,36,24,0,1,30,150,240,120,0,1,62,540,1560,
%T A131689 1800,720,0,1,126,1806,8400,16800,15120,5040,0,1,254,5796,40824,126000,
%U A131689 191520,141120,40320,0,1,510,18150,186480,834120,1905120,2328480
%N A131689 Triangle of numbers T(n,k)=k!*Stirling2(n,k)=A000142(k)*A048993(n,k) 
               read by rows (n>=0,0<=k<=n).
%C A131689 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,0,2,0,3,0,4,0,5,
               0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is 
               the operator defined in A084938 ; another version of A019538 .
%C A131689 See also A019538 : version with n>0 and k>0. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2008]
%F A131689 T(n,k)=k!*(T(n-1,k-1)+T(n-1,k)) with T(0,0)=1. Sum_{k, 0<=k<=n}T(n,k)*x^k 
               = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), 
               A032033(n), A094417(n), A094418(n), A094419(n) for x=-2, -1, 0, 1, 
               2, 3, 4, 5, 6 respectively .
%F A131689 Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)=A000012(n), A000142(n), A000670(n), A122704(n) 
               for x=-1, 0, 1, 2 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 09 2007
%e A131689 Triangle begins:
%e A131689 1;
%e A131689 0, 1;
%e A131689 0, 1, 2;
%e A131689 0, 1, 6, 6;
%e A131689 0, 1, 14, 36, 24;
%e A131689 0, 1, 30, 150, 240, 120;
%e A131689 0, 1, 62, 540, 1560, 1800, 720;
%e A131689 0, 1, 126, 1806, 8400, 16800, 15120, 5040;
%e A131689 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320;
%e A131689 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880 ;
               ...
%Y A131689 Cf. Diagonals : A000142, A001286, A037960, A037961, A037962, A037963.
%Y A131689 Sequence in context: A089949 A085845 A138106 this_sequence A114329 A101371 
               A154974
%Y A131689 Adjacent sequences: A131686 A131687 A131688 this_sequence A131690 A131691 
               A131692
%K A131689 nonn,tabl
%O A131689 0,6
%A A131689 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2007