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

%I A209764 #5 Mar 30 2012 18:58:15
%S A209764 1,2,2,3,4,4,4,8,14,8,5,14,32,34,16,6,22,62,96,86,32,7,32,108,218,280,
%T A209764 202,64,8,44,174,432,718,760,470,128,9,58,264,778,1584,2194,1992,1066,
%U A209764 256,10,74,382,1304,3142,5360,6382,5048,2390,512,11,92,532
%N A209764 Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.
%C A209764 Row n begins with n and ends with 2^(n-1).
%C A209764 Row sums:  1,4,11,34,101,304,911,2734,... A060925.
%C A209764 Alternating row sums: 1,0,3,2,5,4,7,6,... A060925.
%C A209764 For a discussion and guide to related arrays, see A208510.
%F A209764 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209764 v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,
%F A209764 where u(1,x)=1, v(1,x)=1.
%e A209764 First five rows:
%e A209764 1
%e A209764 2...2
%e A209764 3...4....4
%e A209764 4...8....14...8
%e A209764 5...14...32...34...16
%e A209764 First three polynomials v(n,x): 1, 2 + 2x , 3 + 4x + 4x^2.
%t A209764 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209764 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209764 v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
%t A209764 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209764 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209764 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209764 TableForm[cu]
%t A209764 Flatten[%]    (* A209763 *)
%t A209764 Table[Expand[v[n, x]], {n, 1, z}]
%t A209764 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209764 TableForm[cv]
%t A209764 Flatten[%]    (* A209764 *)
%t A209764 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
%t A209764 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
%t A209764 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
%t A209764 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442*)
%Y A209764 Cf. A209663, A208510.
%K A209764 nonn,tabl
%O A209764 1,2
%A A209764 _Clark Kimberling_, Mar 14 2012