OFFSET
1,2
COMMENTS
The column with index 0 of the standard array is not incorporated in this triangle. (It contains a 1 followed by zeros.)
The truncated Fibonacci sequence is A000045(x)/x-1 = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5+ ...
The composition with the Motzkin sequence is A001006(...) = 1 + x + 4*x^2 + 15*x^3 + 58*x^4 + 229*x^5 + ...
LINKS
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
FORMULA
T(n,m) = m*Sum_{k=m..n} Sum_{i=k..n} binomial(i-1,k-1)*binomial(i,n-i)*Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j)/k, n>0, m<=n.
EXAMPLE
1,
3, 1,
9, 6, 1,
31, 27, 9, 1,
113, 116, 54, 12, 1,
431, 493, 282, 90, 15, 1,
1697, 2098, 1383, 556, 135, 18, 1,
6847, 8975, 6567, 3107, 965, 189, 21, 1
PROG
(Maxima)
T(n, m):=m*sum(sum(binomial(i-1, k-1)*binomial(i, n-i), i, k, n)*sum(binomial(j, 2*j-m-k)*binomial(k, j), j, 0, k)/k, k, m, n);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 11 2011
STATUS
approved