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A122449 Diagonal elements A122445(n+2,n) of the pendular trinomial triangle A122445.

%I A122449 #9 Feb 06 2025 10:23:12
%S A122449 1,2,6,22,83,324,1298,5302,22002,92488,392996,1685232,7283511,
%T A122449 31694460,138746706,610601374,2699835614,11988069480,53433418716,
%U A122449 238986495540,1072250526558,4824638825032,21765895919444,98433111857436
%N A122449 Diagonal elements A122445(n+2,n) of the pendular trinomial triangle A122445.
%H A122449 G. C. Greubel, <a href="/A122449/b122449.txt">Table of n, a(n) for n = 0..1000</a>
%F A122449 G.f.: A(x) = B(x)^2/(1+x -x*B(x)) where B(x) is the g.f. of A122446.
%F A122449 G.f.: 2/(1 -2*x -2*x^2 -2*x^3 +4*x^4 +4*x^5 +(1 +2*x^2 +2*x^3)*f(x)), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - _G. C. Greubel_, Mar 17 2021
%F A122449 D-finite with recurrence -4*(n+3)*(37*n-56)*a(n) +(33*n^2-357*n+1624)*a(n-1) +4*(547*n^2-620*n-554)*a(n-2) +4*(1142*n^2-2566*n-1613)*a(n-3) +16*(180*n^2-588*n+65)*a(n-4) +4*(-331*n^2+1937*n+1076)*a(n-5) +8*(-320*n^2+2107*n-617)*a(n-6) -48*(19*n-13)*(n-7)*a(n-7)=0. - _R. J. Mathar_, Feb 06 2025
%t A122449 f[x_] := Sqrt[1 - 4*x - 4*x^2 + 4*x^4];
%t A122449 CoefficientList[Series[2/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f[x]), {x,0,30}], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A122449 (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/(1+x-x*B);polcoeff(A,n,x)}
%o A122449 (Sage)
%o A122449 def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
%o A122449 def A122449_list(prec):
%o A122449     P.<x> = PowerSeriesRing(QQ, prec)
%o A122449     return P( 2/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f(x)) ).list()
%o A122449 A122449_list(30) # _G. C. Greubel_, Mar 17 2021
%o A122449 (Magma)
%o A122449 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A122449 f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
%o A122449 Coefficients(R!( 2/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f(x)) )); // _G. C. Greubel_, Mar 17 2021
%Y A122449 Cf. A122445, A122446, A122447, A122448, A122450, A122451, A122452.
%K A122449 nonn
%O A122449 0,2
%A A122449 _Paul D. Hanna_, Sep 07 2006