A209996 - OEIS

%I #8 Feb 13 2023 08:59:31

%S 1,1,3,1,5,9,1,5,21,27,1,5,25,81,81,1,5,25,117,297,243,1,5,25,125,513,

%T 1053,729,1,5,25,125,609,2133,3645,2187,1,5,25,125,625,2853,8505,

%U 12393,6561,1,5,25,125,625,3093,12825,32805,41553,19683,1,5,25,125

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209998; see the Formula section.

%C Row n starts with 1, 5, 5^2, 5^3,...,5^floor[(n+1)/2] and ends with 3^(n-1).

%C Denoting the general term by T(n,k), we have T(n,n-1)=A081038.

%C Alternating row sums: A000975 (signed).

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

%F u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,

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

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

%e First five rows:

%e 1

%e 1...3

%e 1...5...9

%e 1...5...21...27

%e 1...5...25...81...81

%e First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 9x^2.

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

%t u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;

%t v[n_, x_] := (x + 1)*u[n - 1, x] + 2 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[%]    (* A209996 *)

%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[%]    (* A209998 *)

%Y Cf. A209998, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 23 2012