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Triangle of coefficients of polynomials v(n,x) jointly generated with A209577; see the Formula section.
3

%I #7 Mar 30 2012 18:58:15

%S 1,3,1,5,3,1,9,8,4,1,15,19,13,5,1,25,41,36,19,6,1,41,84,90,60,26,7,1,

%T 67,165,210,169,92,34,8,1,109,315,465,439,287,133,43,9,1,177,588,990,

%U 1073,818,454,184,53,10,1,287,1079,2043,2502,2178,1405,681,246

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209577; see the Formula section.

%C Alternating row sums: 1,0,2,1,3,2,4,3,5,4,... (A028242).

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

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

%e 3...2....1

%e 5...6....3....1

%e 9...13...10...4...1

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

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

%t u[n_, x_] := x*u[n - 1, 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[%] (* A209577 *)

%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[%] (* A209578 *)

%Y Cf. A209577, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 11 2012