A124790 A generalized Motzkin triangle.

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 3, 4, 3, 2, 1, 0, 6, 9, 6, 5, 2, 1, 0, 15, 21, 15, 12, 6, 3, 1, 0, 36, 51, 36, 30, 15, 9, 3, 1, 0, 91, 127, 91, 76, 40, 25, 10, 4, 1, 0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1

Offset

0,13

Comments

Columns include A005043, A001006, A002026. Row sums are A124791. For even k, column k has g.f. x^k*M(x)^(k/2), where M(x)=2/(1-x+sqrt(1-2x-3x^2)) is the g.f. of A001006. For odd k, column k has g.f. x^k*S(x)*M(x)^floor(k/2), S(x)=(1+x-sqrt(1-2x-3x^2))/(2x(1+x)), the g.f. of A005043.

Links

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

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Formula

Triangle is the product of A124788 and A124305, that is, it is the product of the expansion of (1+x*y)/(1-x^2*y^2-x^3*y^2) and the inverse of the Riordan array (1,x(1-x^2)).

Example

Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 1, 2, 1, 1,
0, 3, 4, 3, 2, 1,
0, 6, 9, 6, 5, 2, 1,
0, 15, 21, 15, 12, 6, 3, 1,
0, 36, 51, 36, 30, 15, 9, 3, 1,
0, 91, 127, 91, 76, 40, 25, 10, 4, 1,
0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1
Production matrix begins
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1
- Paul Barry, Apr 07 2011

Crossrefs

Sequence in context: A318754 A318758 A334192 * A325734 A351322 A305736

Adjacent sequences: A124787 A124788 A124789 * A124791 A124792 A124793

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 07 2006

Status

approved