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

0,18

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.

Links

Table of n, a(n) for n=0..104.

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.

Example

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,...

Prog

(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)); ); )

Crossrefs

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

Sequence in context: A014081 A091890 A029431 * A284264 A273302 A025086

Adjacent sequences: A091489 A091490 A091491 * A091493 A091494 A091495

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 16 2004

Status

approved