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A207612 Triangle of coefficients of polynomials u(n,x) jointly generated with A207613; see the Formula section.

%I A207612 #16 May 21 2021 08:09:09
%S A207612 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 A207612 184,160,96,64,88,218,356,440,432,352,192,128,143,402,728,1000,1104,
%U A207612 992,768,384,256,232,730,1452,2184,2656,2688,2240,1664,768,512
%N A207612 Triangle of coefficients of polynomials u(n,x) jointly generated with A207613; see the Formula section.
%C A207612 Column 1:  A000071
%C A207612 Column 2:  2*A023610
%F A207612 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 A207612 where u(1,x)=1, v(1,x)=1.
%e A207612 First five rows:
%e A207612 1
%e A207612 2
%e A207612 4....2
%e A207612 7....6....4
%e A207612 12...14...12...8
%t A207612 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207612 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207612 v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1
%t A207612 Table[Factor[u[n, x]], {n, 1, z}]
%t A207612 Table[Factor[v[n, x]], {n, 1, z}]
%t A207612 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207612 TableForm[cu]
%t A207612 Flatten[%]    (* A207612 *)
%t A207612 Table[Expand[v[n, x]], {n, 1, z}]
%t A207612 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207612 TableForm[cv]
%t A207612 Flatten[%]    (* A207613 *)
%o A207612 (Python)
%o A207612 from sympy import Poly
%o A207612 from sympy.abc import x
%o A207612 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A207612 def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x) + 1
%o A207612 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
%o A207612 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017
%Y A207612 Cf. A207613.
%K A207612 nonn,tabf
%O A207612 1,2
%A A207612 _Clark Kimberling_, Feb 19 2012