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A122450 Diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.

%I A122450 #6 Mar 17 2021 15:32:42
%S A122450 1,3,12,47,189,773,3208,13478,57222,245134,1058348,4600571,20118753,
%T A122450 88450897,390721560,1733348234,7719287578,34497374034,154659735720,
%U A122450 695397289078,3135087583426,14168892518258,64181607367952
%N A122450 Diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.
%H A122450 G. C. Greubel, <a href="/A122450/b122450.txt">Table of n, a(n) for n = 0..1000</a>
%F A122450 G.f.: A(x) = B(x)*(B(x)-1)/(x*(1+x -x*B(x))) where B(x) is the g.f. of A122446.
%F A122450 G.f.: 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - _G. C. Greubel_, Mar 17 2021
%t A122450 f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
%t A122450 CoefficientList[Series[2*(1-2*x^2-f[x])/(x*(1+2*x^2+f[x])*(1-x+2*x^2+2*x^3+(1+x)*f[x])), {x,0,30}], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A122450 (PARI) {a(n)=local(A,B=2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x^2*O(x^n)))); A=B*(B-1)/x/(1+x-x*B);polcoeff(A,n,x)}
%o A122450 (Sage)
%o A122450 def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
%o A122450 def A122449_list(prec):
%o A122450     P.<x> = PowerSeriesRing(QQ, prec)
%o A122450     return P( 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))) ).list()
%o A122450 A122449_list(30) # _G. C. Greubel_, Mar 17 2021
%o A122450 (Magma)
%o A122450 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A122450 f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
%o A122450 Coefficients(R!( 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))) )); // _G. C. Greubel_, Mar 17 2021
%Y A122450 Cf. A122445, A122446, A122447, A122448, A122449, A122451, A122452.
%K A122450 nonn
%O A122450 0,2
%A A122450 _Paul D. Hanna_, Sep 07 2006