|
%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
|
