A093010 Triangle, read by rows, such that the convolution of the n-th row with the natural numbers forms the n-th diagonal, for n>=0, where each row begins with 1.

1, 1, 2, 1, 4, 3, 1, 6, 7, 4, 1, 8, 14, 10, 5, 1, 10, 22, 22, 13, 6, 1, 12, 33, 40, 30, 16, 7, 1, 14, 45, 66, 58, 38, 19, 8, 1, 16, 60, 100, 104, 76, 46, 22, 9, 1, 18, 76, 146, 168, 142, 94, 54, 25, 10, 1, 20, 95, 202, 262, 242, 180, 112, 62, 28, 11, 1, 22, 115, 272, 386, 394, 316

Offset: 0

Author

Paul D. Hanna, Mar 14 2004

Keywords

• nonn
• tabl

Comments

Row sums form A000713, the number of partitions of n into parts of 3 kinds. Antidiagonal sums form A000990, the number of 2-line partitions of n.

Examples

			T(7,3) = 66 = 1*4+8*3+14*2+10*1 = T(4,0)*4+T(4,1)*3+T(4,2)*2+T(4,3)*1; this is also the third term of the 4th-diagonal.
The 6th antidiagonal is {1,10,14,4}, which has a sum of 29 = A000990(6) = number of 2-line partitions of 6.
Rows begin:
{1},
{1,2},
{1,4,3},
{1,6,7,4},
{1,8,14,10,5},
{1,10,22,22,13,6},
{1,12,33,40,30,16,7},
{1,14,45,66,58,38,19,8},
{1,16,60,100,104,76,46,22,9},
{1,18,76,146,168,142,94,54,25,10},
{1,20,95,202,262,242,180,112,62,28,11},
{1,22,115,272,386,394,316,218,130,70,31,12},...
		

Crossrefs

Cf. A000713, A000990, A092905.

Programs

PARI

T(n,k)=if(n
				

Formula

T(n, k) = sum_{j=0..k} (k-j+1)*T(n-k, j), with T(0, n) = 1 for all n>=0.

A000713(n) = sum_{k=0..n} T(n, k) (row sums).

A000990(n) = sum_{k=0..floor(n/2)} T(n-k, k) (antidiagonal sums).