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
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).
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