A227528 - OEIS

%I #31 Jul 05 2017 21:11:05

%S 4,60,840,13860,270270,6126120,158722200,4633467300,150587687250,

%T 5394582443250,211240491462000,8977720887135000,411608985890602500,

%U 20251162105817643000,1064311075116860571000,59509669251792738478500,3527387653231263127556250,220942735734212754080568750

%N a(n) = numerator of r(n) = 2*(3*n+2)!/((2*n)!*2^n), n>=0.

%C The first values with denominators > 1 occur at n = {84, 148, 164, 168, 169, 276, 292, 296, 297, ...}. - _G. C. Greubel_, Jul 04 2017

%H G. C. Greubel, <a href="/A227528/b227528.txt">Table of n, a(n) for n = 0..370</a>

%F In Maple notation,

%F E.g.f. of r: ((135/8)*z+4)*cos((2/3)*arcsin((3/4)*sqrt(6)*sqrt(z)))/(1-(27/8)*z)^2+(3/8)*sqrt(z)*((135/4)*z+17)*sin((2/3)*arcsin((3/4)*sqrt(6)*sqrt(z)))*sqrt(6)/(1-(27/8)*z)^(5/2).

%F E.g.f. of r: 4*hypergeom([4/3,5/3],[1/2],27*z/8).

%F Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),

%F   w(x)=(8/3)*sqrt(3)*((1/36) *(128*x^2/81-40*x/3+20) *exp(-4*x/27) *BesselK(1/3,(4/27)*x)/Pi +(2/81)*x *(-5+16*x/9) *exp(-4*x/27) *BesselK(4/3,4*x/27)/Pi);

%F   w(x)=-(14/243)*16^(2/3)*x^(2/3)*3 *hypergeom([13/6], [4/3], -(8/27)*x)/GAMMA(2/3)-(10/243)*sqrt(3)*16^(1/3)*x^(1/3)*9*GAMMA(2/3)*hypergeom([11/6], [2/3], -(8/27)*x)/Pi.

%F   For x>3.32, w(x)>0.

%F   w(0)=w(3.32)=limit(w(x),x=infinity)=0.

%F   For x<3.32, w(x)<0.

%F r(n) = int(x^n*w(x), x=0..infinity), n>=0.

%F Asymptotics: r(n)->(1/1152)*sqrt(6)*(10368*n^2+10224*n+2161)*(27/8)^n*exp(-n)*(n^n), for n->infinity.

%F The rational values are given by 4*(-2*n+1)*r(n) + 3*(3*n+2)*(3*n+1) * r(n-1)=0. - _R. J. Mathar_, Jul 20 2013

%p seq(numer(4*(3*n+2)!/((2*n)!*2^(n+1))), n=0..14);

%t Table[Numerator[2(3n + 2)!/((2n)! 2^n)], {n, 0, 84}]  (* _G. C. Greubel_, Jul 04 2017 *)

%o (PARI) for(n=0,50, print1(numerator(2*(3*n + 2)!/((2*n)!*2^n)), ", ")) \\ _G. C. Greubel_, Jul 04 2017

%Y Cf. A001879.

%K nonn,frac

%O 0,1

%A _Karol A. Penson_, Jul 14 2013

%E Erroneous definition corrected by _G. C. Greubel_, Jul 04 2017