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A061162 a(n) = (6n)!n!/((3n)!(2n)!^2). 9
1, 30, 2310, 204204, 19122246, 1848483780, 182327718300, 18236779032600, 1842826521244230, 187679234340049620, 19232182592635611060, 1980665038436368775400, 204826599735691440534300, 21255328931341321610645544, 2212241139727064219063537016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.

This sequence is the particular case a = 3, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and gcd(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A211419 (a = 3, b = 2), A211420 (a = 4, b = 1) and A211421 (a = 4, b = 3) and A061163 (a = 5, b = 1). The o.g.f. Sum_{n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012

REFERENCES

M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..100

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], J. London Math. Soc., Vol. 79, Issue 2, (2009) 422-444.

W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.

F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

FORMULA

a(n) ~ 1/2*Pi^(-1/2)*n^(-1/2)*2^(2*n)*3^(3*n)*{1 - 1/72*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002

n*(2*n-1)*a(n) -6*(6*n-1)*(6*n-5)*a(n-1)=0. - R. J. Mathar, Oct 26 2014

From Peter Bala, Aug 21 2016: (Start)

a(n) = Sum_{k = 0..2*n} binomial(6*n, k)*binomial(4*n - k - 1, 2*n - k).

a(n) = Sum_{k = 0..n} binomial(8*n, 2*n - 2*k)*binomial(2*n + k - 1, k).

O.g.f. A(x) = Hypergeom([5/6, 1/6], [1/2], 108*x).

a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^6/(1 - x)^2. Cf. A091496 and A262732. It follows that the o.g.f. A(x) for this sequence is the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.

Let G(x) = 1/x * series reversion( x*(1 - x)/(1 + x)^3 ) = 1 + 4*x + 23*x^2 + 156*x^3 + 1162*x^4 + ..., essentially the o.g.f. for A007297. Then A(x^2) equals the even part of 1 + x*d/dx(Log(G(x)).

exp(Sum_{n >= 1} a(n)*x^n/n) = F(x), where F(x) = 1 + 30*x + 1605*x^2 + 107218*x^3 + 8043114*x^4 + 647773116*x^5 + 54730094637*x^6 + ... has integer coefficients since F(x^2) = G(x)*G(-x). Furthermore, F(x)^(1/6) = 1 + 5*x + 205*x^2 + 12328*x^3 + 874444*x^4 + 68022261*x^5 + 5613007167*x^6 + ... appears to have all integer coefficients. (End)

a(n) is the n-th moment of the positive weight function w(x) on x = (0,108), i.e., in Maple notation: a(n) = int(x^n*w(x), x=0..108), n=0,1,..., where w(x) = sqrt(3)*(1 + sqrt(1 - x/108))^(2/3)/(12*2^(1/3)*Pi*x^(5/6)*sqrt(1 - x/108)) + 2^(4/3) *sqrt(3)/(864*Pi*x^(1/6)*(1 + sqrt(1 - x/108))^(2/3)*sqrt(1 - x/108)). The weight function w(x) is singular at x=0 and at x=108 and is the solution of the Hausdorff moment problem. This solution is unique. - Karol A. Penson, Mar 01 2018

MAPLE

A061162 := n->(6*n)!*n!/((3*n)!*(2*n)!^2);

MATHEMATICA

a[n_] := 16^n Gamma[3 n + 1/2]/(Gamma[n + 1/2] Gamma[2 n + 1]);

Table[a[n], {n, 0, 14}] (* Peter Luschny, Mar 01 2018 *)

PROG

(PARI) { for (n=0, 100, write("b061162.txt", n, " ", (6*n)!*n!/((3*n)!*(2*n)!^2)) ) } \\ Harry J. Smith, Jul 18 2009

CROSSREFS

Cf. A061163, A061164, A007297, A091527, A262732.

See also A091527, A211419, A211420, A211421.

Sequence in context: A056070 A301377 A054647 * A230272 A230612 A138916

Adjacent sequences:  A061159 A061160 A061161 * A061163 A061164 A061165

KEYWORD

easy,nonn

AUTHOR

Richard Stanley, Apr 17 2001

STATUS

approved

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Last modified April 1 02:00 EDT 2020. Contains 333153 sequences. (Running on oeis4.)