A102230 Triangle, read by rows, where each column equals the convolution of A032349 with the prior column, starting with column 0 equal to A032349 shift right.

1, 1, 1, 4, 5, 1, 24, 32, 9, 1, 172, 236, 76, 13, 1, 1360, 1896, 656, 136, 17, 1, 11444, 16116, 5828, 1348, 212, 21, 1, 100520, 142544, 53112, 13184, 2376, 304, 25, 1, 911068, 1298524, 494364, 128924, 25436, 3804, 412, 29, 1, 8457504, 12100952

Offset: 0

Author

Paul D. Hanna, Jan 01 2005

Keywords

• nonn
• tabl

Comments

Row sums equal A027307; the self-convolution of the row sums form A032349. Column 0 equals A032349 shift right. Column 1 is A102231. This triangle is a variant of A100326.

Examples

			This triangle is generated by the recurrence:
T(n,k) = Sum_{i=0..n-k} T(i+1,0)*T(n-i-1,k-1) for n>k>0,
T(n,0) = Sum_{i=0..n-1} (2*i+1)*T(n-1,i) for n>0, with T(0,0)=1.
Rows begin:
[1],
[1,1],
[4,5,1],
[24,32,9,1],
[172,236,76,13,1],
[1360,1896,656,136,17,1],
[11444,16116,5828,1348,212,21,1],
[100520,142544,53112,13184,2376,304,25,1],...
Column 0 is formed from the partial sums of the prior row
after a term-by-term product with the odd numbers:
T(2,0) = 1*T(1,0) + 3*T(1,1) = 1*1 + 3*1 = 4.
T(3,0) = 1*T(2,0) + 3*T(2,1) + 5*T(2,2) = 1*4 + 3*5 + 5*1 = 24.
		

Crossrefs

Cf. A032349, A027307, A102231, A100326.

Programs

PARI

{T(n,k)=if(n
				

Formula

G.f.: A(x, y) = (1+x*F(x))/(1-x*y*F(x)) where F(x) is the g.f. of A032349 and satisfies F(x) = (1+x*F(x))^2/(1-x*F(x))^2.