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Triangle of coefficients of polynomials u(n,x) jointly generated with A210865; see the Formula section.
3

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

%S 1,2,1,3,3,2,4,9,9,3,5,21,30,21,5,6,41,91,96,47,8,7,71,242,358,278,

%T 101,13,8,113,565,1187,1303,757,209,21,9,169,1182,3517,5238,4364,1951,

%U 422,34,10,241,2263,9332,18816,21213,13674,4802,833,55,11,331

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210865; see the Formula section.

%C Row n starts with n and ends with F(n), where F=A000045 (Fibonacci numbers).

%C Column 2: A064999

%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...1

%e 3...3....2

%e 4...9....9....3

%e 5...21...30...21...5

%e First three polynomials u(n,x): 1, 2 + x, 3 + 3x + 2x^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. A210865, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 28 2012