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A122451 A diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.

%I A122451 #6 Mar 17 2021 15:32:49
%S A122451 1,4,17,72,305,1300,5576,24068,104510,456332,2002675,8829892,39096653,
%T A122451 173781548,775183764,3469084436,15571135682,70084045640,316242702258,
%U A122451 1430351652352,6483550388522,29448610671464,134010580021152
%N A122451 A diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.
%H A122451 G. C. Greubel, <a href="/A122451/b122451.txt">Table of n, a(n) for n = 0..1000</a>
%F A122451 G.f.: A(x) = B(x)^2*(B(x)-1)/(x*(1+x - x*B(x))) where B(x) is the g.f. of A122446.
%F A122451 G.f.: 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(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 A122451 f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
%t A122451 CoefficientList[Series[4*(1-2*x^2-f[x])/(x*(1+2*x^2+f[x])^2*(1-x+2*x^2+2*x^3+(1+x)*f[x])), {x,0,30}], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A122451 (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^2*(B-1)/x/(1+x-x*B);polcoeff(A,n,x)}
%o A122451 (Sage)
%o A122451 def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
%o A122451 def A122449_list(prec):
%o A122451     P.<x> = PowerSeriesRing(QQ, prec)
%o A122451     return P( 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))) ).list()
%o A122451 A122449_list(30) # _G. C. Greubel_, Mar 17 2021
%o A122451 (Magma)
%o A122451 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A122451 f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
%o A122451 Coefficients(R!( 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))) )); // _G. C. Greubel_, Mar 17 2021
%Y A122451 Cf. A122445, A122446, A122447, A122448, A122449, A122450, A122452.
%K A122451 nonn
%O A122451 0,2
%A A122451 _Paul D. Hanna_, Sep 07 2006