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A209763 Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section.

%I A209763 #6 Mar 30 2012 18:58:15
%S A209763 1,1,2,2,5,4,3,9,13,8,4,15,31,35,16,5,23,61,97,85,32,6,33,107,219,279,
%T A209763 203,64,7,45,173,433,717,761,469,128,8,59,263,779,1583,2195,1991,1067,
%U A209763 256,9,75,381,1305,3141,5361,6381,5049,2389,512,10,93,531
%N A209763 Triangle of coefficients of polynomials u(n,x) jointly generated with A209764; see the Formula section.
%C A209763 Row n begins with n and ends with 2^(n-1).
%C A209763 Row sums:  1,3,11,33,101,303,911,... A081250
%C A209763 Alternating row sums: 1,-1,1,-1,1,.. A033999
%C A209763 For a discussion and guide to related arrays, see A208510.
%F A209763 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209763 v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,
%F A209763 where u(1,x)=1, v(1,x)=1.
%e A209763 First five rows:
%e A209763 1
%e A209763 1...2
%e A209763 2...5....4
%e A209763 3...9....13...8
%e A209763 4...15...31...35...16
%e A209763 First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
%t A209763 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209763 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209763 v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
%t A209763 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209763 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209763 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209763 TableForm[cu]
%t A209763 Flatten[%]    (* A209763 *)
%t A209763 Table[Expand[v[n, x]], {n, 1, z}]
%t A209763 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209763 TableForm[cv]
%t A209763 Flatten[%]    (* A209764 *)
%t A209763 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
%t A209763 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
%t A209763 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
%t A209763 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442 *)
%Y A209763 Cf. A209764, A208510.
%K A209763 nonn,tabl
%O A209763 1,3
%A A209763 _Clark Kimberling_, Mar 14 2012