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A122448 Diagonal elements A122445(n+1,n) of the pendular trinomial triangle A122445.

%I A122448 #9 Feb 06 2025 10:23:20
%S A122448 1,1,3,10,36,135,525,2094,8524,35266,147862,626884,2682940,11575707,
%T A122448 50295809,219879814,966487380,4268781902,18936044682,84326759820,
%U A122448 376853237480,1689551241606,7597003401186,34251504489484
%N A122448 Diagonal elements A122445(n+1,n) of the pendular trinomial triangle A122445.
%H A122448 G. C. Greubel, <a href="/A122448/b122448.txt">Table of n, a(n) for n = 0..1000</a>
%F A122448 G.f.: A(x) = B(x)/(1 +x -x*B(x) ) where B(x) is the g.f. of A122446.
%F A122448 G.f. satisfies: A(x) = 1 + x*(1-2*x-2*x^2)*A(x) + x^2*(4+3*x)*A(x)^2.
%F A122448 G.f.: A(x) = 2/(1 -x +2*x^2 +2*x^3 + (1+x)*sqrt(1 -4*x -4*x^2 +4*x^4)).
%F A122448 D-finite with recurrence 4*(n+2)*a(n) +(-9*n-2)*a(n-1) +(-41*n+34)*a(n-2) +2*(-20*n+39)*a(n-3) +4*(n-7)*a(n-4) +4*(7*n-36)*a(n-5) +12*(n-6)*a(n-6)=0. - _R. J. Mathar_, Feb 06 2025
%t A122448 f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
%t A122448 CoefficientList[Series[2/(1-x+2*x^2+2*x^3 +(1+x)*f[x]), {x,0,30}], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A122448 (PARI) {a(n) =polcoeff(2/(1-x+2*x^2+2*x^3 +(1+x)*sqrt(1-4*x-4*x^2+4*x^4 +x*O(x^n) )), n)}
%o A122448 (Sage)
%o A122448 def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
%o A122448 def A122447_list(prec):
%o A122448     P.<x> = PowerSeriesRing(QQ, prec)
%o A122448     return P( 2/(1-x+2*x^2+2*x^3 +(1+x)*f(x)) ).list()
%o A122448 A122447_list(30) # _G. C. Greubel_, Mar 17 2021
%o A122448 (Magma)
%o A122448 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A122448 f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
%o A122448 Coefficients(R!( 2/(1-x+2*x^2+2*x^3 +(1+x)*f(x)) )); // _G. C. Greubel_, Mar 17 2021
%Y A122448 Cf. A122445, A122446, A122447, A122449, A122450, A122451, A122452.
%K A122448 nonn
%O A122448 0,3
%A A122448 _Paul D. Hanna_, Sep 07 2006