login
Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} j!*C(n-1,j-1)*T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.
1

%I #6 Apr 29 2015 06:39:00

%S 1,1,1,1,3,1,1,11,7,1,1,49,47,13,1,1,261,341,139,21,1,1,1631,2731,

%T 1471,329,31,1,1,11743,24173,16213,4789,671,43,1,1,95901,235463,

%U 189373,69441,12881,1231,57,1,1,876809,2509621,2357503,1032245,237961,30169,2087,73,1

%N Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} j!*C(n-1,j-1)*T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.

%C Can be understood as a convolution matrix or as a sequence-to-triangle transformation similar to the partial Bell polynomials defined as: S -> T(n, k, S) = Sum_{j=0..n-k+1} C(n-1,j-1)*S(j)*T(n-j,k-1,S) if k != 0 else S(0)^n. Here S(n) = n!. The case S(n) = n gives the triangle of idempotent numbers A059297 and the case S(n) = 1 for all n leads to A256894.

%F T(n+1,1) = A001339(n) for n>=0.

%F T(n,n-1) = A002061(n) for n>=1.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 11, 7, 1;

%e 1, 49, 47, 13, 1;

%e 1, 261, 341, 139, 21, 1;

%p # Implemented as a sequence transformation acting on f: n -> n!.

%p F := proc(n, k, f) option remember; `if`(k=0, f(0)^n,

%p add(binomial(n-1, j-1)*f(j)*F(n-j, k-1, f), j=0..n-k+1)) end:

%p for n from 0 to 7 do seq(F(n, k, j->j!), k=0..n) od;

%Y Cf. A001339, A002061, A059297, A256894.

%K nonn,tabl,easy

%O 0,5

%A _Peter Luschny_, Apr 28 2015