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Triangle of coefficients of polynomials v(n,x) jointly generated with A207616; see the Formula section.
3

%I #9 Mar 31 2023 03:22:21

%S 1,2,1,3,3,4,7,1,5,14,5,6,25,16,1,7,41,41,7,8,63,91,29,1,9,92,182,92,

%T 9,10,129,336,246,46,1,11,175,582,582,175,11,12,231,957,1254,550,67,1,

%U 13,298,1507,2508,1507,298,13,14,377,2288,4719,3718,1079,92,1

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A207616; see the Formula section.

%F u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=xu(n-1,x)+v(n-1,x)+1, where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 2...1

%e 3...3

%e 4...7...1

%e 5...14...5

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

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

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

%t Table[Factor[u[n, x]], {n, 1, z}]

%t Table[Factor[v[n, x]], {n, 1, z}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A207616 *)

%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[%] (* A207617 *)

%Y Cf. A207616.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Feb 20 2012