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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.

%I A102230 #8 Jul 19 2024 19:05:11
%S A102230 1,1,1,4,5,1,24,32,9,1,172,236,76,13,1,1360,1896,656,136,17,1,11444,
%T A102230 16116,5828,1348,212,21,1,100520,142544,53112,13184,2376,304,25,1,
%U A102230 911068,1298524,494364,128924,25436,3804,412,29,1,8457504,12100952
%N 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.
%C A102230 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 A102230 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 A102230 This triangle is generated by the recurrence:
%e A102230 T(n,k) = Sum_{i=0..n-k} T(i+1,0)*T(n-i-1,k-1) for n>k>0,
%e A102230 T(n,0) = Sum_{i=0..n-1} (2*i+1)*T(n-1,i) for n>0, with T(0,0)=1.
%e A102230 Rows begin:
%e A102230 [1],
%e A102230 [1,1],
%e A102230 [4,5,1],
%e A102230 [24,32,9,1],
%e A102230 [172,236,76,13,1],
%e A102230 [1360,1896,656,136,17,1],
%e A102230 [11444,16116,5828,1348,212,21,1],
%e A102230 [100520,142544,53112,13184,2376,304,25,1],...
%e A102230 Column 0 is formed from the partial sums of the prior row
%e A102230 after a term-by-term product with the odd numbers:
%e A102230 T(2,0) = 1*T(1,0) + 3*T(1,1) = 1*1 + 3*1 = 4.
%e A102230 T(3,0) = 1*T(2,0) + 3*T(2,1) + 5*T(2,2) = 1*4 + 3*5 + 5*1 = 24.
%o A102230 (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 A102230 Cf. A032349, A027307, A102231, A100326.
%K A102230 nonn,tabl
%O A102230 0,4
%A A102230 _Paul D. Hanna_, Jan 01 2005