A071264 - OEIS

%I #13 Sep 13 2024 08:01:24

%S 1,4,16,63,244,934,3549,13428,50694,191216,721259,2722073,10282610,

%T 38886330,147243585,558279000,2119599270,8058312360,30677134380,

%U 116937340494,446317079712,1705563385196,6525404350306,24994411455928

%N Expansion of (1+x*C^4)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

%H G. C. Greubel, <a href="/A071264/b071264.txt">Table of n, a(n) for n = 0..1000</a>

%F Conjecture: D-finite with recurrence (n+6) *(8587*n^2 +62958*n +61667) *a(n) +(8587*n^3 -377960*n^2 -2268967*n -1091596) *a(n-1) -2 *(2*n-1) *(42935*n^2 -14497*n -123334) *a(n-2)=0. - _R. J. Mathar_, Dec 10 2015

%p g := (1-sqrt(1-4*x))/2/x ;

%p (1+x*g^4)*g^3 ;

%p taylor(%,x=0,40) ;

%p gfun[seriestolist](%) ; # _R. J. Mathar_, Aug 14 2024

%t CoefficientList[Series[(1-7*x+14*x^2-6*x^3-3*x^4 -(1-5*x+6*x^2-x^4 )*Sqrt[1-4*x])/(2*x^6), {x,0,30}], x] (* _G. C. Greubel_, Sep 12 2024 *)

%o (Magma)

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

%o p:= func< x | (1 -7*x +14*x^2 -6*x^3 -3*x^4 -(1 -5*x +6*x^2 -x^4)*Sqrt(1-4*x))/2 >;

%o Coefficients(R!( p(x) )); // _G. C. Greubel_, Sep 12 2024

%o (SageMath)

%o def p(x): return (1-7*x+14*x^2-6*x^3-3*x^4 -(1-5*x+6*x^2 -x^4)*sqrt(1-4*x))/(2*x^6)

%o def A071264_list(prec):

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

%o     return P( p(x) ).list()

%o A071264_list(30) # _G. C. Greubel_, Sep 12 2024

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jun 06 2002