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A099512 Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + 3*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

%I A099512 #5 Jun 13 2017 22:13:13
%S A099512 1,1,3,1,6,1,1,9,11,6,1,12,30,45,1,1,15,58,144,30,9,1,18,95,330,195,
%T A099512 144,1,1,21,141,630,685,873,58,12,1,24,196,1071,1770,3258,685,330,1,1,
%U A099512 27,260,1680,3801,9198,3989,3258,95,15,1,30,333,2484,7210,21672,15533
%N A099512 Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + 3*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.
%C A099512 Row sums form A099513. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).
%F A099512 G.f.: (1-x+3*x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2-7*x^3*y^2+x^4*y^4).
%e A099512 Rows begin:
%e A099512 [1],
%e A099512 [1,3],
%e A099512 [1,6,1],
%e A099512 [1,9,11,6],
%e A099512 [1,12,30,45,1],
%e A099512 [1,15,58,144,30,9],
%e A099512 [1,18,95,330,195,144,1],
%e A099512 [1,21,141,630,685,873,58,12],
%e A099512 [1,24,196,1071,1770,3258,685,330,1],
%e A099512 [1,27,260,1680,3801,9198,3989,3258,95,15],...
%e A099512 and can be derived from coefficients of (1+3*z+z^2)^n:
%e A099512 [1],
%e A099512 [1,3,1],
%e A099512 [1,6,11,6,1],
%e A099512 [1,9,30,45,30,9,1],
%e A099512 [1,12,58,144,195,144,58,12,1],
%e A099512 [1,15,95,330,685,873,685,330,95,15,1],...
%e A099512 by shifting each column k down by [k/2] rows.
%o A099512 (PARI) T(n,k)=if(n<k || k<0,0,polcoeff((1+3*z+z^2+z*O(z^k))^(n-k\2),k,z))
%Y A099512 Cf. A099509, A099510, A099513.
%K A099512 nonn,tabl
%O A099512 0,3
%A A099512 _Paul D. Hanna_, Oct 20 2004