A114202 A Pascal-Jacobsthal triangle.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 16, 16, 5, 1, 1, 6, 27, 42, 27, 6, 1, 1, 7, 41, 87, 87, 41, 7, 1, 1, 8, 58, 156, 216, 156, 58, 8, 1, 1, 9, 78, 254, 456, 456, 254, 78, 9, 1, 1, 10, 101, 386, 860, 1122, 860, 386, 101, 10, 1

Offset: 0

Author

Paul Barry, Nov 16 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A114203. T(2n,n) is A114204. Inverse has row sums 0^n.

Examples

			Triangle begins
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1;
  1, 4,  8,  4,  1;
  1, 5, 16, 16,  5,  1;
  1, 6, 27, 42, 27,  6, 1;
  1, 7, 41, 87, 87, 41, 7, 1;
  ...
		

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, with J(n) = A001045(n):

T(n, k) = Sum_{i=0..n-k} C(n-k, i)*C(k, i)*J(i);

T(n, k) = Sum_{i=0..n} C(n-k, n-i)*C(k, i-k)*J(i-k);

T(n, k) = Sum_{i=0..n} C(k, i)*C(n-k, n-i)*J(k-i) if k <= n, and 0 otherwise.

As a square array, with J(n) = A001045(n):

T(n, k) = Sum_{i=0..n} C(n, i)C(k, i)*J(i);

T(n, k) = Sum_{i=0..n+k} C(n, n+k-i)*C(k, i-k)*J(i-k);

Column k has g.f. (Sum_{i=0..k} C(k, i)*J(i+1)*(x/(1 - x))^i)*x^k/(1 - x).