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Triangle of coefficients of polynomials v(n,x) jointly generated with A210864; see the Formula section.
3

%I #6 Mar 31 2012 20:30:51

%S 1,2,2,6,7,3,12,21,18,5,20,61,75,42,8,30,151,262,231,93,13,42,323,829,

%T 1025,656,196,21,56,617,2330,3935,3607,1742,401,34,72,1081,5815,13578,

%U 16849,11723,4380,799,55,90,1771,13070,42167,69475,65727

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210864; see the Formula section.

%C For n>1, row n starts with n(n-1) and ends with F(n+1), where F=A000045 (Fibonacci numbers).

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

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

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

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

%e First five rows:

%e 1

%e 2....2

%e 6....7....3

%e 12...21...18...5

%e 20...61...75...34...8

%e First three polynomials v(n,x): 1, 2 + 2x, 6 + 7x + 3x^2

%t u[1, x_] := 1; v[1, x_] := 1; z = 14;

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

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

%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[%] (* A210864 *)

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

%t TableForm[cv]

%t Flatten[%] (* A210865 *)

%Y Cf. A210864, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 28 2012