0,18

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.

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.

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,...

(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)); ); )}

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

Adjacent sequences: A091489 A091490 A091491 this_sequence A091493 A091494 A091495

Sequence in context: A014081 A091890 A029431 this_sequence A025086 A035699 A132406

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 16 2004