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Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.
3

%I #6 Jul 12 2012 00:40:00

%S 1,3,1,4,3,2,5,6,8,3,6,10,18,14,5,7,15,33,38,27,8,8,21,54,81,83,49,13,

%T 9,28,82,150,197,170,89,21,10,36,118,253,401,448,342,159,34,11,45,163,

%U 399,736,999,987,671,282,55,12,55,218,598,1253,1988,2387,2106

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.

%C For n>1, row n starts with n+1, followed by the n-th

%C triangular number, and ends in F(n+1), where F=A000045

%C (Fibonacci numbers).

%C Column 3: A166830.

%C Row sums: A048654.

%C Alternating row sums: 1,2,3,4,5,6,7,8,9,...

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

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

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

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

%e First five rows:

%e 1

%e 3...1

%e 4...3....2

%e 5...6....8....3

%e 6...10...18...14...5

%e First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 2x^2.

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

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

%t v[n_, x_] := (x + 1)*u[n - 1, x] + 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[%] (* A209703 *)

%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[%] (* A209704 *)

%Y Cf. A209703, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 12 2012