Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #8 Jul 19 2024 19:05:11
%S 1,1,1,4,5,1,24,32,9,1,172,236,76,13,1,1360,1896,656,136,17,1,11444,
%T 16116,5828,1348,212,21,1,100520,142544,53112,13184,2376,304,25,1,
%U 911068,1298524,494364,128924,25436,3804,412,29,1,8457504,12100952
%N 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.
%C 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.
%F 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.
%e This triangle is generated by the recurrence:
%e T(n,k) = Sum_{i=0..n-k} T(i+1,0)*T(n-i-1,k-1) for n>k>0,
%e T(n,0) = Sum_{i=0..n-1} (2*i+1)*T(n-1,i) for n>0, with T(0,0)=1.
%e Rows begin:
%e [1],
%e [1,1],
%e [4,5,1],
%e [24,32,9,1],
%e [172,236,76,13,1],
%e [1360,1896,656,136,17,1],
%e [11444,16116,5828,1348,212,21,1],
%e [100520,142544,53112,13184,2376,304,25,1],...
%e Column 0 is formed from the partial sums of the prior row
%e after a term-by-term product with the odd numbers:
%e T(2,0) = 1*T(1,0) + 3*T(1,1) = 1*1 + 3*1 = 4.
%e T(3,0) = 1*T(2,0) + 3*T(2,1) + 5*T(2,2) = 1*4 + 3*5 + 5*1 = 24.
%o (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)));))}
%Y Cf. A032349, A027307, A102231, A100326.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Jan 01 2005