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A257566 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} P(n,j)*T(n-j,k-1) if k>0 else 0^n where P(n,j) is the number of j-partitions of n; for n>=0 and 0<=k<=n.

%I A257566 #8 Apr 30 2015 21:12:47
%S A257566 1,0,1,0,2,1,0,3,4,1,0,5,13,7,1,0,7,30,30,10,1,0,11,76,119,65,14,1,0,
%T A257566 15,152,357,306,113,18,1,0,22,330,1119,1375,746,193,23,1,0,30,633,
%U A257566 2973,5059,3888,1497,295,28,1,0,42,1245,8036,18605,19423,10298,2930,447,34,1
%N A257566 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} P(n,j)*T(n-j,k-1) if k>0 else 0^n where P(n,j) is the number of j-partitions of n; for n>=0 and 0<=k<=n.
%F A257566 T(n,1) = A000041(n) for n>=1.
%F A257566 T(n,n-1) = A014616(n-1) for n>=2.
%e A257566 1;
%e A257566 0,  1;
%e A257566 0,  2,  1;
%e A257566 0,  3,  4,   1;
%e A257566 0,  5, 13,   7,  1;
%e A257566 0,  7, 30,  30, 10,  1;
%e A257566 0, 11, 76, 119, 65, 14, 1;
%p A257566 T := proc(n,k) option remember; `if`(k=0, 0^n,
%p A257566 add(combinat:-numbpart(n,j)*T(n-j, k-1), j=0..n-k+1)) end:
%p A257566 for n from 0 to 12 do seq(T(n, k), k=0..n) od;
%Y A257566 Cf. A000041, A014616.
%K A257566 nonn,tabl,easy
%O A257566 0,5
%A A257566 _Peter Luschny_, Apr 30 2015