%I #19 Feb 13 2023 05:52:16
%S 1,4,1,14,8,1,46,44,12,1,141,204,90,16,1,409,846,538,152,20,1,1132,
%T 3234,2787,1112,230,24,1,3011,11600,13035,6892,1990,324,28,1,7736,
%U 39502,56372,37956,14345,3236,434,32,1,19275,128765,228921,191008,90749,26586,4914,560,36,1,46724,404228,882291,894364,519580,189798,45311,7088,702,40,1
%N Triangle T(n,r), read by rows, where the r-th column is expansion of A(x)^r, with A(x) = x * (x+1) * (2*x^4+4*x^3-2*x+1) * (x^4+2*x^3-x+1) / (x^2+x-1)^6.
%C f(x) = (x+x^2)/(1-x-x^2), g(x)=x+x^2, h(x)=x+x^2+x^3, A(x)=g(h(f(x))).
%C A(x)^r = sum(n>=r, T(n,r)*x^n) and composition G(A(x)) = g(0)+sum(n>0, sum(r=1..n, T(n,r)*g(r))*x^n).
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%F T(n,r) = sum(m=r..n, sum(k=m..n,sum(i=k..n, binomial(i,n-i)*binomial(i-1,k-1)) *sum(j=0..m, binomial(m,j) *binomial(j,k-3*m+2*j))) *binomial(r,m-r)), n>0, 1<=r<=n.
%e Triangle begins:
%e 1;
%e 4,1;
%e 14,8,1;
%e 46,44,12,1;
%e 141,204,90,16,1;
%e 409,846,538,152,20,1;
%e 1132,3234,2787,1112,230,24,1;
%e 3011,11600,13035,6892,1990,324,28,1;
%o (Maxima)
%o T(n,r):= sum(sum(sum(binomial(i,n-i)*binomial(i-1,k-1),i,k,n) *sum(binomial(m,j) *binomial(j,k-3*m+2*j),j,0,m),k,m,n) *binomial(r,m-r),m,r,n);
%K nonn,tabl
%O 1,2
%A _Vladimir Kruchinin_, Mar 02 2011