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
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.
Programs
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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.
Comments