cp's OEIS Frontend

A209996 Triangle of coefficients of polynomials u(n,x) jointly generated with A209998; see the Formula section.

%I A209996 #8 Feb 13 2023 08:59:31
%S A209996 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 A209996 1053,729,1,5,25,125,609,2133,3645,2187,1,5,25,125,625,2853,8505,
%U A209996 12393,6561,1,5,25,125,625,3093,12825,32805,41553,19683,1,5,25,125
%N A209996 Triangle of coefficients of polynomials u(n,x) jointly generated with A209998; see the Formula section.
%C A209996 Row n starts with 1, 5, 5^2, 5^3,...,5^floor[(n+1)/2] and ends with 3^(n-1).
%C A209996 Denoting the general term by T(n,k), we have T(n,n-1)=A081038.
%C A209996 Alternating row sums: A000975 (signed).
%C A209996 For a discussion and guide to related arrays, see A208510.
%F A209996 u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A209996 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A209996 where u(1,x)=1, v(1,x)=1.
%e A209996 First five rows:
%e A209996 1
%e A209996 1...3
%e A209996 1...5...9
%e A209996 1...5...21...27
%e A209996 1...5...25...81...81
%e A209996 First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 9x^2.
%t A209996 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209996 u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209996 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209996 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209996 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209996 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209996 TableForm[cu]
%t A209996 Flatten[%]    (* A209996 *)
%t A209996 Table[Expand[v[n, x]], {n, 1, z}]
%t A209996 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209996 TableForm[cv]
%t A209996 Flatten[%]    (* A209998 *)
%Y A209996 Cf. A209998, A208510.
%K A209996 nonn,tabl
%O A209996 1,3
%A A209996 _Clark Kimberling_, Mar 23 2012