A122451 - OEIS

%I #6 Mar 17 2021 15:32:49

%S 1,4,17,72,305,1300,5576,24068,104510,456332,2002675,8829892,39096653,

%T 173781548,775183764,3469084436,15571135682,70084045640,316242702258,

%U 1430351652352,6483550388522,29448610671464,134010580021152

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

%H G. C. Greubel, <a href="/A122451/b122451.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 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 f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];

%t 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 (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 (Sage)

%o def f(x): return sqrt(1-4*x-4*x^2+4*x^4)

%o def A122449_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     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 A122449_list(30) # _G. C. Greubel_, Mar 17 2021

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;

%o 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 Cf. A122445, A122446, A122447, A122448, A122449, A122450, A122452.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 07 2006