A124790 - OEIS

%I #11 Feb 21 2014 03:19:56

%S 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,

%T 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,

%U 323,232,196,105,69,29,14,4,1

%N A generalized Motzkin triangle.

%C 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.

%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

%F 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)).

%e Triangle begins

%e 1,

%e 0, 1,

%e 0, 0, 1,

%e 0, 1, 1, 1,

%e 0, 1, 2, 1, 1,

%e 0, 3, 4, 3, 2, 1,

%e 0, 6, 9, 6, 5, 2, 1,

%e 0, 15, 21, 15, 12, 6, 3, 1,

%e 0, 36, 51, 36, 30, 15, 9, 3, 1,

%e 0, 91, 127, 91, 76, 40, 25, 10, 4, 1,

%e 0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1

%e Production matrix begins

%e 0, 1,

%e 0, 0, 1,

%e 0, 1, 1, 1,

%e 0, 0, 0, 0, 1,

%e 0, 1, 1, 1, 1, 1,

%e 0, 0, 0, 0, 0, 0, 1,

%e 0, 1, 1, 1, 1, 1, 1, 1,

%e 0, 0, 0, 0, 0, 0, 0, 0, 1,

%e 0, 1, 1, 1, 1, 1, 1, 1, 1, 1

%e - _Paul Barry_, Apr 07 2011

%K easy,nonn,tabl

%O 0,13

%A _Paul Barry_, Nov 07 2006