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A091492 Triangle, read by rows, generated recursively and related to partitions.

%I A091492 #7 Jun 13 2017 21:51:27
%S A091492 1,1,1,1,1,0,1,1,1,0,1,1,1,0,0,1,1,2,0,0,0,1,1,2,1,0,0,0,1,1,3,1,0,0,
%T A091492 0,0,1,1,3,2,0,0,0,0,0,1,1,4,3,0,0,0,0,0,0,1,1,4,4,1,0,0,0,0,0,0,1,1,
%U A091492 5,5,1,1,0,0,0,0,0,0,1,1,5,7,2,1,0,0,0,0,0,0,0,1,1,6,8,3,2,0,0,0,0,0,0,0,0
%N A091492 Triangle, read by rows, generated recursively and related to partitions.
%C A091492 Excluding the leading zeros, the columns are related to partitions. The 3rd column lists A001399 (partitions of n into at most 3 parts). The 4th column lists A001400 (partitions of n into at most 4 parts). The 5th column lists A001401 (partitions of n into at most 5 parts). The 6th column is A091498. Row sums are A091493. The number of nonzero terms in each row is A091497.
%F A091492 T(n, k)=Sum T(n-k, j)*T(j, k-j) {j=[(k+1)/2]..min(k, n-k)}, with T(0, 0)=1, T(n, 0)=1, T(1, 1)=1.
%e A091492 T(12,3) = 7 = (4)*1+(3)*1 = T(9,2)*T(2,1)+T(9,3)*T(3,0) = Sum T(9,j)*T(j,3-j) {j=2..3}.
%e A091492 Rows begin:
%e A091492 {1},
%e A091492 {1,1},
%e A091492 {1,1,0},
%e A091492 {1,1,1,0},
%e A091492 {1,1,1,0,0},
%e A091492 {1,1,2,0,0,0},
%e A091492 {1,1,2,1,0,0,0},
%e A091492 {1,1,3,1,0,0,0,0},
%e A091492 {1,1,3,2,0,0,0,0,0},
%e A091492 {1,1,4,3,0,0,0,0,0,0},
%e A091492 {1,1,4,4,1,0,0,0,0,0,0},
%e A091492 {1,1,5,5,1,1,0,0,0,0,0,0},
%e A091492 {1,1,5,7,2,1,0,0,0,0,0,0,0},
%e A091492 {1,1,6,8,3,2,0,0,0,0,0,0,0,0},
%e A091492 {1,1,6,10,5,3,0,0,0,0,0,...
%e A091492 {1,1,7,12,6,5,0,0,0,0,0,...
%e A091492 {1,1,7,14,9,7,1,0,0,0,0,...
%e A091492 {1,1,8,16,11,10,2,0,0,0,...
%e A091492 {1,1,8,19,15,13,3,2,0,0,...
%e A091492 {1,1,9,21,18,18,5,2,0,0,...
%e A091492 {1,1,9,24,23,23,8,4,0,0,...
%e A091492 {1,1,10,27,27,30,11,6,0,...
%e A091492 {1,1,10,30,34,37,17,10,0,...
%o A091492 (PARI) T(n,k)=if(k>n || n<0 || k<0,0,if(k<=1 || (k==n && n<2),1, sum(j=(k+1)\2,min(n-k,k),T(n-k,j)*T(j,k-j)););)
%Y A091492 Cf. A001399, A001400, A001401, A091493, A091497, A091498.
%K A091492 nonn,tabl
%O A091492 0,18
%A A091492 _Paul D. Hanna_, Jan 16 2004