

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, 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
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OFFSET

0,4


COMMENTS

Row sums equal A027307; the selfconvolution 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))/(1x*y*F(x)) where F(x) is the g.f. of A032349 and satisfies F(x) = (1+x*F(x))^2/(1x*F(x))^2.


EXAMPLE

This triangle is generated by the recurrence:
T(n,k) = Sum_{i=0..nk} T(i+1,0)*T(ni1,k1) for n>k>0,
T(n,0) = Sum_{i=0..n1} (2*i+1)*T(n1,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 termbyterm 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<kk<0, 0, if(n==0, 1, if(k==0, sum(i=0, n1, (2*i+1)*T(n1, i)), sum(i=0, nk, T(i+1, 0)*T(ni1, k1))); ))}


CROSSREFS

Cf. A032349, A027307, A102231, A100326.
Sequence in context: A210590 A108446 A109962 * A147724 A110519 A204579
Adjacent sequences: A102227 A102228 A102229 * A102231 A102232 A102233


KEYWORD

nonn,tabl


AUTHOR

Paul D. Hanna, Jan 01 2005


STATUS

approved



