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

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, 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, 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

Offset: 0

Author

Paul D. Hanna, Jan 16 2004

Keywords

• nonn
• tabl

Comments

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.

Examples

			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}.
Rows begin:
{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},
{1,1,3,1,0,0,0,0},
{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},
{1,1,5,5,1,1,0,0,0,0,0,0},
{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},
{1,1,6,10,5,3,0,0,0,0,0,...
{1,1,7,12,6,5,0,0,0,0,0,...
{1,1,7,14,9,7,1,0,0,0,0,...
{1,1,8,16,11,10,2,0,0,0,...
{1,1,8,19,15,13,3,2,0,0,...
{1,1,9,21,18,18,5,2,0,0,...
{1,1,9,24,23,23,8,4,0,0,...
{1,1,10,27,27,30,11,6,0,...
{1,1,10,30,34,37,17,10,0,...
		

Crossrefs

Cf. A001399, A001400, A001401, A091493, A091497, A091498.

Programs

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)););)

Formula

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.