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A210756 Triangle of coefficients of polynomials v(n,x) jointly generated with A210755; see the Formula section.

%I A210756 #5 Mar 30 2012 18:58:17
%S A210756 1,3,2,6,10,5,10,30,33,12,15,70,127,100,29,21,140,371,472,291,70,28,
%T A210756 252,910,1656,1624,822,169,36,420,1974,4800,6640,5294,2273,408,45,660,
%U A210756 3906,12144,22166,24702,16589,6184,985,55,990,7194,27720,63954
%N A210756 Triangle of coefficients of polynomials v(n,x) jointly generated with A210755; see the Formula section.
%C A210756 Column 1: triangular numbers, A000217
%C A210756 Coefficient of v(n,x):  A000129(n)
%C A210756 Row sums:  A002450
%C A210756 Alternating row sums: 1,1,1,1,1,1,1,1,1,...
%C A210756 For a discussion and guide to related arrays, see A208510.
%F A210756 u(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
%F A210756 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210756 where u(1,x)=1, v(1,x)=1.
%e A210756 First five rows:
%e A210756 1
%e A210756 3....2
%e A210756 6....10...5
%e A210756 10...30...33....12
%e A210756 15...70...127...100...29
%e A210756 First three polynomials v(n,x): 1, 3 + 2x, 6 + 10x + 5x^2
%t A210756 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210756 u[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A210756 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210756 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210756 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210756 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210756 TableForm[cu]
%t A210756 Flatten[%]    (* A210755 *)
%t A210756 Table[Expand[v[n, x]], {n, 1, z}]
%t A210756 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210756 TableForm[cv]
%t A210756 Flatten[%]    (* A210756 *)
%t A210756 Table[u[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)
%t A210756 Table[v[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)
%Y A210756 Cf. A210755, A208510.
%K A210756 nonn,tabl
%O A210756 1,2
%A A210756 _Clark Kimberling_, Mar 25 2012