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Triangle of coefficients of polynomials v(n,x) jointly generated with A210755; see the Formula section.
3

%I #5 Mar 30 2012 18:58:17

%S 1,3,2,6,10,5,10,30,33,12,15,70,127,100,29,21,140,371,472,291,70,28,

%T 252,910,1656,1624,822,169,36,420,1974,4800,6640,5294,2273,408,45,660,

%U 3906,12144,22166,24702,16589,6184,985,55,990,7194,27720,63954

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

%C Column 1: triangular numbers, A000217

%C Coefficient of v(n,x): A000129(n)

%C Row sums: A002450

%C Alternating row sums: 1,1,1,1,1,1,1,1,1,...

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,

%F v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 3....2

%e 6....10...5

%e 10...30...33....12

%e 15...70...127...100...29

%e First three polynomials v(n,x): 1, 3 + 2x, 6 + 10x + 5x^2

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;

%t v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A210755 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210756 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)

%Y Cf. A210755, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 25 2012