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

0,5

Comments

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

Links

Table of n, a(n) for n=0..65.

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).

Example

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;
  ...

Crossrefs

Sequence in context: A073134 A300260 A026692 * A125806 A347148 A202756

Adjacent sequences: A114199 A114200 A114201 * A114203 A114204 A114205

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 16 2005

Status

approved