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A275607 a(n) = 2*12^n*Gamma(n+1/2)*(n+1)/(sqrt(Pi)*Gamma(n+3)). 1
1, 4, 27, 216, 1890, 17496, 168399, 1667952, 16888014, 173997720, 1818276174, 19225409616, 205299909828, 2210922105840, 23984556773175, 261854925711840, 2874948871877910, 31722346066169880, 351589335566716170, 3912422681494285200, 43694647856506630620, 489597172255515289680 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In reference of K. Szymanski et al. the function g(x) from the Eq.(4.6) satisfies the equality g(x/4)/4 = W(x) where W(x) is the weight function of the integral representation, see below.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..925

Simeon T. Stefanov, Counting fixed points free vector fields on B^2, arXiv:1807.03714 [math.GT], 2018.

K. Szymanski, B. Collins, T. Szarek and K. Zyczkowski, Convex set of quantum states with positive partial transpose analysed by hit and run algorithm, arXiv:1611.01194 [quant-ph], 2016.

FORMULA

O.g.f: (1/54)*(1-(6*z+1)*sqrt(1-12*z))/z^2;

E.g.f.(in Maple notation): (1/9)*exp(6*z)*(6*z*(BesselI(0,6*z)-BesselI(1,6*z))+ BesselI(1,6*z))/z;

Recurrence: (-12*n^2-54*n-54)*a(n+1)+(n^2+6*n+8)*a(n+2)=0, n=0,1..., for the initial values a(0)=1, a(1)=4.

Integral representation as the n-th Hausdorff moment of the positive function W(x) on the segment x=(0,12), i.e., in Maple notation, a(n)= int(x^n*W(x),x=0..12), where W(x)=(1/27)*sqrt(12-x)*(3+(1/2)*x)/(Pi*sqrt(x)). This representation is unique.

a(n) ~ 2^(2*n+1)*3^n/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Nov 14 2016

a(n) = 2*3^n*binomial(2n+1, n-1)*(n+1)/(2n^2+n). - Charles R Greathouse IV, Nov 14 2016

MAPLE

a := n -> (2^(2*n+1)*3^n*(n+1)*GAMMA(n+1/2))/(sqrt(Pi)*GAMMA(n+3)):

seq(a(n), n=0..21); # Peter Luschny, Nov 14 2016

MATHEMATICA

g[z_] :=  E^z (BesselI[0, z] - (1-1/z) BesselI[1, z])

Table[CoefficientList[2/3 Series[g[6z], {z, 0, 21}], z]] Range[0, 21]! //Flatten (* Peter Luschny, Nov 14 2016 *)

Table[ 2*12^n*(n + 1)*Gamma[n + 1/2]/(Sqrt[Pi]*Gamma[n + 3]), {n, 0, 100}] (* G. C. Greubel, Jan 13 2017 *)

PROG

(PARI) a(n)=2*12^n*gamma(n+1/2)*(n+1)\/(sqrt(Pi)*(n+2)!) \\ Charles R Greathouse IV, Nov 14 2016

(PARI) a(n)=2*3^n*binomial(2*n+1, n-1)*(n+1)/(2*n+1)/n \\ Charles R Greathouse IV, Nov 14 2016

CROSSREFS

Cf. A000108, A002293, A002894, A000168, A000139, A000257, A004987, A005568, A000888, A004981.

Sequence in context: A026005 A059391 A190738 * A319518 A304045 A317103

Adjacent sequences:  A275604 A275605 A275606 * A275608 A275609 A275610

KEYWORD

nonn

AUTHOR

Karol A. Penson, Nov 14 2016

STATUS

approved

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Last modified June 6 14:04 EDT 2020. Contains 334827 sequences. (Running on oeis4.)