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Triangle of coefficients of polynomials u(n,x) jointly generated with A209704; see the Formula section.
3

%I #5 Mar 30 2012 18:58:15

%S 1,0,2,0,3,3,0,4,6,5,0,5,10,14,8,0,6,15,28,28,13,0,7,21,48,66,55,21,0,

%T 8,28,75,129,149,104,34,0,9,36,110,225,326,319,193,55,0,10,45,154,363,

%U 626,774,661,352,89,0,11,55,208,553,1099,1625,1761,1332,634

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209704; see the Formula section.

%C For n>1, row n begins with 0, has second term n, and ends with F(n+1), where F=A000045 (Fibonacci numbers); for n>2, column 2 consists of triangular numbers.

%C Row sums: A098790.

%C Alternating row sums: 1,-2,0,-3,-1,-4,-3,-5,-3,-6,,...

%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 0...2

%e 0...3....3

%e 0...4....6...5

%e 0...5...10...14...8

%e First three polynomials v(n,x): 1, 2x, 3x + 3x^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. A209704, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 12 2012