cp's OEIS Frontend

A119376 Second diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

%I A119376 #12 Jan 25 2025 02:13:06
%S A119376 1,4,16,63,248,980,3894,15563,62555,252789,1026623,4188390,17159382,
%T A119376 70570380,291253664,1205935204,5008047097,20854723702,87064706122,
%U A119376 364334839028,1527943938306,6420911995109,27033938458595
%N A119376 Second diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.
%C A119376 Equals convolution of A119370 and A119375, which is the prior diagonal above the central terms of triangle A119369.
%H A119376 G. C. Greubel, <a href="/A119376/b119376.txt">Table of n, a(n) for n = 0..1000</a>
%F A119376 G.f.: A(x) = B(x)^2*(G(x) - 1)/x^2 = B(x)^2*(B(x) - 1)/(x+x^2 - x^2*B(x)), where B(x) is g.f. of A119370 and G(x) is g.f. of A119371 (central terms of A119369).
%F A119376 G.f.: 2*(1-2*x-x^2-f(x))/( x^2*(1+2*x^3+x^4+(1+x)^2*f(x))*(1+x^2+f(x)) ) where f(x) = sqrt(1-4*x-2*x^2+x^4). - _G. C. Greubel_, Mar 17 2021
%t A119376 f[x_]:= Sqrt[1-4*x-2*x^2+x^4];
%t A119376 CoefficientList[Series[2*(1-2*x-x^2 -f[x])/(x^2*(1+2*x^3+x^4 +(1+x)^2*f[x])*(1+x^2 +f[x])), {x,0,30}], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A119376 (PARI) {a(n)=polcoeff(4/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))^2* (2*(1+x)/(1+4*x+x^2 + sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x^3*O(x^n)))-1)/x^2,n)}
%o A119376 (SageMath)
%o A119376 def f(x): return sqrt(1-4*x-2*x^2+x^4)
%o A119376 def A119376_list(prec):
%o A119376     P.<x> = PowerSeriesRing(QQ, prec)
%o A119376     return P( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) ).list()
%o A119376 A119376_list(30) # _G. C. Greubel_, Mar 17 2021
%o A119376 (Magma)
%o A119376 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A119376 f:= func< x | Sqrt(1-4*x-2*x^2+x^4) >;
%o A119376 Coefficients(R!( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) )); // _G. C. Greubel_, Mar 17 2021
%Y A119376 Cf. A119369, A119370, A119371, A119372, A119373, A119374, A119375.
%K A119376 nonn
%O A119376 0,2
%A A119376 _Paul D. Hanna_, May 17 2006