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Triangle of coefficients of polynomials u(n,x) jointly generated with A207613; see the Formula section.
3

%I #16 May 21 2021 08:09:09

%S 1,2,4,2,7,6,4,12,14,12,8,20,30,32,24,16,33,60,76,72,48,32,54,116,168,

%T 184,160,96,64,88,218,356,440,432,352,192,128,143,402,728,1000,1104,

%U 992,768,384,256,232,730,1452,2184,2656,2688,2240,1664,768,512

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A207613; see the Formula section.

%C Column 1: A000071

%C Column 2: 2*A023610

%F u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 2

%e 4....2

%e 7....6....4

%e 12...14...12...8

%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] + 2 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[%] (* A207612 *)

%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[%] (* A207613 *)

%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) + 2*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. A207613.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Feb 19 2012