OFFSET
1,2
COMMENTS
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))).
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).
LINKS
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
FORMULA
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.
EXAMPLE
Triangle begins:
1;
4,1;
14,8,1;
46,44,12,1;
141,204,90,16,1;
409,846,538,152,20,1;
1132,3234,2787,1112,230,24,1;
3011,11600,13035,6892,1990,324,28,1;
PROG
(Maxima)
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);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 02 2011
STATUS
approved