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A210794 Triangle of coefficients of polynomials v(n,x) jointly generated with A210793; see the Formula section.

%I A210794 #10 Sep 08 2013 19:59:33
%S A210794 1,1,2,3,3,3,3,11,8,5,9,18,29,17,8,9,48,67,71,35,13,27,81,180,194,158,
%T A210794 68,21,27,189,387,575,508,338,129,34,81,324,918,1410,1617,1222,695,
%U A210794 239,55,81,702,1890,3606,4471,4222,2793,1393,436,89,243,1215
%N A210794 Triangle of coefficients of polynomials v(n,x) jointly generated with A210793; see the Formula section.
%C A210794 Row n starts with a power of 3 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
%C A210794 Row sums: A000244 (powers of 3)
%C A210794 Alternating row sums: A077925
%C A210794 For a discussion and guide to related arrays, see A208510.
%F A210794 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A210794 v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),
%F A210794 where u(1,x)=1, v(1,x)=1.
%F A210794 T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 29 2012
%e A210794 First five rows:
%e A210794 1
%e A210794 1...2
%e A210794 3...3....3
%e A210794 3...11...8....5
%e A210794 9...18...29...17...8
%e A210794 First three polynomials v(n,x): 1, 1 + 2x, 3 + 3x + 3x^2
%t A210794 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210794 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210794 d[x_] := h + x; e[x_] := p + x;
%t A210794 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210794 j = 1; c = 0; h = 2; p = -1; f = 0;
%t A210794 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210794 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210794 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210794 TableForm[cu]
%t A210794 Flatten[%]   (* A210793 *)
%t A210794 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210794 TableForm[cv]
%t A210794 Flatten[%]   (* A210794 *)
%t A210794 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
%t A210794 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
%t A210794 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
%t A210794 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
%Y A210794 Cf. A210793, A208510.
%K A210794 nonn,tabl
%O A210794 1,3
%A A210794 _Clark Kimberling_, Mar 26 2012