A114202 - OEIS

%I #12 Jan 29 2019 07:22:35

%S 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,

%T 87,87,41,7,1,1,8,58,156,216,156,58,8,1,1,9,78,254,456,456,254,78,9,1,

%U 1,10,101,386,860,1122,860,386,101,10,1

%N A Pascal-Jacobsthal triangle.

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

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%F As a number triangle, with J(n) = A001045(n):

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

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

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

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

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

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

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

%e Triangle begins

%e   1;

%e   1, 1;

%e   1, 2,  1;

%e   1, 3,  3,  1;

%e   1, 4,  8,  4,  1;

%e   1, 5, 16, 16,  5,  1;

%e   1, 6, 27, 42, 27,  6, 1;

%e   1, 7, 41, 87, 87, 41, 7, 1;

%e   ...

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Nov 16 2005