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%I #7 Mar 17 2021 08:02:39
%S 1,2,6,20,70,253,938,3546,13617,52967,208255,826315,3304456,13304924,
%T 53891402,219442686,897772983,3688451380,15211545938,62950542636,
%U 261329456566,1087985751336,4541524025769,19003488710465,79696345430789
%N A lower diagonal of pendular trinomial triangle A119369.
%H G. C. Greubel, <a href="/A119373/b119373.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: A(x) = B(x)^2/(1+x - x*B(x)) = B(x)^2*G(x) = B(x)*H(x), where B(x) is g.f. of A119370, G(x) is g.f. of A119371 and H(x) is g.f. of A119372.
%F G.f.: 8*(1+x)/( ((1+x^2) +sqrt((1+x^2)^2 -4*x*(1+x)))^2*(1+4*x+x^2 +sqrt((1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x))) ).
%t CoefficientList[Series[8*(1+x)/( ((1+x^2) + Sqrt[(1+x^2)^2 -4*x*(1+x)])^2*(1 + 4*x +x^2 +Sqrt[(1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x)])), {x,0,30}], x] (* _G. C. Greubel_, Mar 16 2021 *)
%o (PARI) {a(n)=polcoeff(8*(1+x)/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x*O(x^n)))^2 /(1+4*x+x^2 + sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x)+x*O(x^n))),n)}
%o (Sage)
%o def A119373_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( 8*(1+x)/( ((1+x^2) +sqrt((1+x^2)^2 -4*x*(1+x)))^2*(1+4*x+x^2 +sqrt((1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x))) ) ).list()
%o A119373_list(30) # _G. C. Greubel_, Mar 16 2021
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 30);
%o Coefficients(R!( 8*(1+x)/( ((1+x^2) +Sqrt((1+x^2)^2 -4*x*(1+x)))^2*(1+4*x+x^2 +Sqrt((1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x))) ) )); // _G. C. Greubel_, Mar 16 2021
%Y Cf. A119369, A119370, A119371, A119372, A119374, A119375, A119376.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 17 2006
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