A130521 Triangle, read by rows, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>=k>=2, with T(n+1,1) = T(n+1,0) = T(n,n) and T(0,0) = 1 for n>=0.

1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 8, 11, 11, 11, 15, 19, 25, 33, 33, 33, 44, 55, 70, 89, 114, 114, 114, 147, 180, 224, 279, 349, 438, 438, 438, 552, 666, 813, 993, 1217, 1496, 1845, 1845, 1845, 2283, 2721, 3273, 3939, 4752, 5745, 6962, 8458, 8458, 8458, 10303

Offset

0,6

Comments

G.f. of column 0 (A127782) satisfies: G(x) = 1 + x*G(x+x^2).

Links

Table of n, a(n) for n=0..57.

Formula

T(n,0) = Sum_{k=0..[n/2]} C(n-k,k)*T(n-k-1,0) for n>0 with T(0,0)=1. For column 1, T(n,1) = Sum_{k=0..[n/2]+1} [C(n-k,k) + C(n-k+1,k-1)]*T(n-k-1,1) for n>=2, with T(0,1)=T(1,1)=1.

Example

T(5,3) = T(5,2) + T(4,1) = 15 + 4 = 19;
T(6,4) = T(6,3) + T(5,2) = 55 + 15 = 70;
T(7,0) = T(6,6) = 89 + 25 = 114.
Triangle begins:
1;
1, 1;
1, 1, 2;
2, 2, 3, 4;
4, 4, 6, 8, 11;
11, 11, 15, 19, 25, 33;
33, 33, 44, 55, 70, 89, 114;
114, 114, 147, 180, 224, 279, 349, 438;
438, 438, 552, 666, 813, 993, 1217, 1496, 1845;
1845, 1845, 2283, 2721, 3273, 3939, 4752, 5745, 6962, 8458; ...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, if(n==0, 1, T(n-1, n-1)), T(n, k-1)+T(n-1, k-2)))

Crossrefs

Cf. A127782 (column 0), A130522 (diagonal).

Sequence in context: A029076 A259774 A036015 * A369574 A365720 A332251

Adjacent sequences: A130518 A130519 A130520 * A130522 A130523 A130524

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 02 2007

Status

approved