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Triangle of coefficients of polynomials u(n,x) jointly generated with A207611; see the Formula section.
3

%I #14 Apr 13 2020 09:32:48

%S 1,2,4,1,7,3,1,12,7,3,1,20,15,8,3,1,33,30,19,9,3,1,54,58,42,23,10,3,1,

%T 88,109,89,55,27,11,3,1,143,201,182,125,69,31,12,3,1,232,365,363,273,

%U 166,84,35,13,3,1,376,655,709,579,383,212,100,39,14,3,1,609

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A207611; see the Formula section.

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

%e First five rows:

%e 1

%e 2

%e 4...1

%e 7...3...1

%e 12...7...3...1

%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_] := u[n - 1, x] + 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[%] (* A207610 *)

%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[%] (* A207611 *)

%o (Python)

%o from sympy import Poly

%o from sympy.abc import x

%o def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)

%o def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1

%o def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]

%o for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017

%Y Cf. A207611.

%Y Cf. A000071 (column 1), A023610 (column 2).

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Feb 19 2012