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A210801 Triangle of coefficients of polynomials u(n,x) jointly generated with A210802; see the Formula section.

%I A210801 #5 Mar 30 2012 18:58:17
%S A210801 1,3,1,6,5,2,12,15,10,3,21,39,37,19,5,39,90,111,81,35,8,66,198,300,
%T A210801 281,171,64,13,120,414,750,855,659,346,115,21,201,846,1776,2391,2230,
%U A210801 1474,684,205,34,363,1683,4044,6255,6828,5441,3170,1323,362,55
%N A210801 Triangle of coefficients of polynomials u(n,x) jointly generated with A210802; see the Formula section.
%C A210801 Row n ends with F(n), where F=A000045 (Fibonacci numbers).
%C A210801 Row sums:  A003462
%C A210801 Alternating row sums:  1,2,3,4,5,6,7,8,...
%C A210801 For a discussion and guide to related arrays, see A208510.
%F A210801 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210801 v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x)+1,
%F A210801 where u(1,x)=1, v(1,x)=1.
%e A210801 First five rows:
%e A210801 1
%e A210801 3....1
%e A210801 6....5....2
%e A210801 12...15...10...3
%e A210801 21...39...37...19...5
%e A210801 First three polynomials u(n,x): 1, 3 + x, 6 + 5x + 2x^2.
%t A210801 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210801 u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
%t A210801 d[x_] := h + x; e[x_] := p + x;
%t A210801 v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
%t A210801 j = 1; c = 1; h = 2; p = -1; f = 1;
%t A210801 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210801 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210801 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210801 TableForm[cu]
%t A210801 Flatten[%]   (* A210801 *)
%t A210801 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210801 TableForm[cv]
%t A210801 Flatten[%]   (* A210802 *)
%t A210801 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A003462 *)
%t A210801 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A003462 *)
%t A210801 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)
%t A210801 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077898 *)
%Y A210801 Cf. A210802, A208510.
%K A210801 nonn,tabl
%O A210801 1,2
%A A210801 _Clark Kimberling_, Mar 27 2012