

A093010


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


2



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
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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 2line partitions of n.


LINKS



FORMULA

T(n, k) = sum_{j=0..k} (kj+1)*T(nk, 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(nk, 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 4thdiagonal.
The 6th antidiagonal is {1,10,14,4}, which has a sum of 29 = A000990(6) = number of 2line 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, nk), (kj+1)*T(nk, j))))


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



