A119375 - OEIS

%I #7 Mar 17 2021 08:02:53

%S 1,3,11,40,149,564,2166,8420,33074,131085,523599,2105727,8519469,

%T 34652696,141621164,581266730,2394961851,9902433681,41074316737,

%U 170869972460,712729001716,2980264528670,12490379959184,52458339164169

%N Diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

%H G. C. Greubel, <a href="/A119375/b119375.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = B(x)*(G(x) - 1)/x^2 = B(x)*(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 G.f.: (1-2*x-x^2 -sqrt(1-4*x-2*x^2+x^4))/( x^2*(1+2*x^3+x^4 +(1+x)^2*sqrt(1-4*x-2*x^2+x^4)) ). - _G. C. Greubel_, Mar 16 2021

%t CoefficientList[Series[(1-2*x-x^2 -Sqrt[1-4*x-2*x^2+x^4])/(x^2*(1+2*x^3+x^4 +(1+x)^2*Sqrt[1-4*x-2*x^2+x^4])), {x,0,30}], x] (* _G. C. Greubel_, Mar 16 2021 *)

%o (PARI) {a(n)=polcoeff(2/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))* (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 (Sage)

%o def A119375_list(prec):

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

%o     return P( (1-2*x-x^2 -sqrt(1-4*x-2*x^2+x^4))/( x^2*(1+2*x^3+x^4 +(1+x)^2*sqrt(1-4*x-2*x^2+x^4)) ) ).list()

%o A119375_list(30) # _G. C. Greubel_, Mar 16 2021

%o (Magma)

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

%o Coefficients(R!( (1-2*x-x^2 - Sqrt(1-4*x-2*x^2+x^4))/( 1+2*x^3+x^4 +(1+x)^2*Sqrt(1-4*x-2*x^2+x^4) ) )); // _G. C. Greubel_, Mar 16 2021

%Y Cf. A119369, A119370, A119371, A119372, A119373, A119374, A119376.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 17 2006