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A122447 Central terms of pendular trinomial triangle A122445.

%I A122447 #6 Mar 17 2021 15:32:19
%S A122447 1,0,1,2,8,28,107,418,1676,6848,28418,119444,507440,2175500,9400207,
%T A122447 40895602,178984212,787503168,3481278734,15454765948,68871993872,
%U A122447 307981243608,1381569997998,6215433403188,28036071086296
%N A122447 Central terms of pendular trinomial triangle A122445.
%C A122447 G.f.: A(x) = 1/(1+x - x*B(x)) = (1 + x*H(x))/(1+x) = 1 + x^2*F(x)/B(x), where B(x) is g.f. of A122446, H(x) is g.f. of A122448, F(x) is g.f. of A122450.
%H A122447 G. C. Greubel, <a href="/A122447/b122447.txt">Table of n, a(n) for n = 0..1000</a>
%F A122447 G.f. satisfies: A(x) = 1+2*x - 2*x*(3+x)*A(x) + x*(4+3*x)*A(x)^2.
%F A122447 G.f.: A(x) = ( 1 +6*x +2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4) )/( 2*x*(4+3*x) ).
%t A122447 f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
%t A122447 CoefficientList[Series[(1+6*x+2*x^2-f[x])/(2*x*(4+3*x)), {x,0,30}], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A122447 (PARI) {a(n)=polcoeff(2*(1+2*x)/(1+6*x+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))),n)}
%o A122447 (Sage)
%o A122447 def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
%o A122447 def A122447_list(prec):
%o A122447     P.<x> = PowerSeriesRing(QQ, prec)
%o A122447     return P( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) ).list()
%o A122447 A122447_list(30) # _G. C. Greubel_, Mar 17 2021
%o A122447 (Magma)
%o A122447 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A122447 f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
%o A122447 Coefficients(R!( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) )); // _G. C. Greubel_, Mar 17 2021
%Y A122447 Cf. A122445, A122446, A122448, A122449, A122450, A122451, A122452.
%K A122447 nonn
%O A122447 0,4
%A A122447 _Paul D. Hanna_, Sep 07 2006