OFFSET
0,5
LINKS
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
FORMULA
As a number triangle, T(n,k) = Sum_{j=0..n-k} C(n-k, j)C(k, j)F(j);
As a number triangle, T(n,k) = Sum_{j=0..n} C(n-k, n-j)C(k, j-k)F(j-k);
As a number triangle, T(n,k) = Sum_{j=0..n} C(k, j)C(n-k, n-j)F(k-j) if k <= n, 0 otherwise.
As a square array, T(n,k) = Sum_{j=0..n} C(n, j)C(k, j)F(j);
As a square array, T(n,k) = Sum_{j=0..n+k} C(n, n+k-j)C(k, j-k)F(j-k);
Column k has g.f.: (Sum_{j=0..k} C(k, j)F(j+1)(x/(1-x))^j)*x^k/(1-x);
G.f.: -((x^2-x)*y-x+1)/((x^4+x^3-x^2)*y^2+(x^3-3*x^2+2*x)*y-x^2+2*x-1). - Vladimir Kruchinin, Jan 15 2018
EXAMPLE
Triangle begins
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 7, 4, 1;
1, 5, 13, 13, 5, 1;
1, 6, 21, 31, 21, 6, 1;
1, 7, 31, 61, 61, 31, 7, 1;
1, 8, 43, 106, 142, 106, 43, 8, 1;
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Nov 16 2005
STATUS
approved