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

0,4

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.

Links

Table of n, a(n) for n=0..46.

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.

Example

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.

Prog

(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==0, 1, if(k==0, sum(i=0, n-1, (2*i+1)*T(n-1, i)), sum(i=0, n-k, T(i+1, 0)*T(n-i-1, k-1))); ))}

Crossrefs

Cf. A032349, A027307, A102231, A100326.

Sequence in context: A108446 A283263 A109962 * A147724 A110519 A286796

Adjacent sequences: A102227 A102228 A102229 * A102231 A102232 A102233

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 01 2005

Status

approved