A071264 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.

1, 4, 16, 63, 244, 934, 3549, 13428, 50694, 191216, 721259, 2722073, 10282610, 38886330, 147243585, 558279000, 2119599270, 8058312360, 30677134380, 116937340494, 446317079712, 1705563385196, 6525404350306, 24994411455928

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
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 >;
Coefficients(R!( p(x) )); // G. C. Greubel, Sep 12 2024

Maple

g := (1-sqrt(1-4*x))/2/x ;
(1+x*g^4)*g^3 ;
taylor(%,x=0,40) ;
gfun[seriestolist](%) ; # R. J. Mathar, Aug 14 2024

Mathematica

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 *)

SageMath

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)
def A071264_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( p(x) ).list()
A071264_list(30) # G. C. Greubel, Sep 12 2024

Formula

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