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

0,3

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.

Links

Table of n, a(n) for n=0..72.

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).

Example

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},...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, sum(j=0, min(k, n-k), (k-j+1)*T(n-k, j))))

Crossrefs

Cf. A000713, A000990, A092905.

Sequence in context: A134543 A305540 A197871 * A179000 A210559 A180803

Adjacent sequences: A093007 A093008 A093009 * A093011 A093012 A093013

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 14 2004

Status

approved