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

%I A209998 #8 Sep 03 2021 13:58:15
%S A209998 1,2,3,2,8,9,2,10,30,27,2,10,46,108,81,2,10,50,198,378,243,2,10,50,
%T A209998 242,810,1296,729,2,10,50,250,1122,3186,4374,2187,2,10,50,250,1234,
%U A209998 4986,12150,14580,6561,2,10,50,250,1250,5946,21330,45198,48114,19683
%N A209998 Triangle of coefficients of polynomials v(n,x) jointly generated with A209996; see the Formula section.
%C A209998 Row n starts 2, 2*5, 2*5^2,... ; ends with 3^(n-1).
%C A209998 Conjecture: penultimate term in row n is A199923(n).
%C A209998 Alternating row sums: A077925
%C A209998 For a discussion and guide to related arrays, see A208510.
%F A209998 u(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A209998 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A209998 where u(1,x)=1, v(1,x)=1.
%e A209998 First five rows:
%e A209998 1
%e A209998 2...3
%e A209998 2...8....9
%e A209998 2...10...30...27
%e A209998 2...10...46...108...81
%e A209998 First three polynomials v(n,x): 1, 2 + 3x , 2 + 8x + 9x^2.
%t A209998 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209998 u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209998 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209998 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209998 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209998 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209998 TableForm[cu]
%t A209998 Flatten[%]    (* A209996 *)
%t A209998 Table[Expand[v[n, x]], {n, 1, z}]
%t A209998 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209998 TableForm[cv]
%t A209998 Flatten[%]    (* A209998 *)
%Y A209998 Cf. A209996, A208510.
%K A209998 nonn,tabl
%O A209998 1,2
%A A209998 _Clark Kimberling_, Mar 23 2012
%E A209998 a(55) corrected by _Georg Fischer_, Sep 03 2021