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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 (list; table; graph; refs; listen; history; text; internal format)
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: 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

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Last modified December 10 13:27 EST 2016. Contains 279004 sequences.