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A347854 a(n) = (6*n)!/((3*n)!*(2*n)!) * (n/2)!/(3*n/2)!. 4
1, 40, 4620, 622336, 89237148, 13236695040, 2005604901300, 308350245273600, 47913489552349980, 7505566011722039296, 1183237138556438547120, 187495217080545878999040, 29836408028165719837829700, 4764790302634058161217077248 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Fractional factorials are defined using the Gamma function; for example, (n/2)! := Gamma(1 + n/2).

The sequence defined by u(n) = (12*n)!*n!/((6*n)!*(4*n)!*(3*n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 1). See A295431. It is known that u(n) is integral and satisfies the congruences u(n*p) == u(n) ( mod p^3 ) for prime p >= 5 and any positive integer n (Zudilin, Section 5); the o.g.f. Sum_{n >= 0} u(n)*x^n is algebraic over Q(x) (Rodriguez-Villegas).

Here we are essentially considering the sequence ( u(n/2) )n>=0. The sequence is conjectured to be integral.

LINKS

Table of n, a(n) for n=0..13.

P. Bala, Some integer ratios of factorials

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

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

Wadim Zudilin, Congruences for q-binomial coefficients, arXiv:1901.07843 [math.NT], 2019.

FORMULA

a(n) = binomial(6*n,2*n)*binomial(4*n,n)/binomial(3*n/2,n).

a(2*n) = A295431(n).

a(2*n) = 24*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)/( n*(2*n - 1)*(3*n - 1)*(3*n - 2) )*a(2*n-2);

a(2*n+1) = 96*(12*n + 1)*(12*n - 1)*(12*n + 5)*(12*n - 5)/( n*(2*n + 1)*(6*n + 1)*(6*n - 1) )*a(2*n-1).

Asymptotics: a(n) ~ 32^n/sqrt(6*Pi*n) * 3^(3*n/2) as n -> infinity.

O.g.f.: A(x) = hypergeom([1/12, 5/12, 7/12, 11/12], [1/3, 1/2, 2/3], 27648*x^2) + 40*x*hypergeom([11/12, 13/12, 7/12, 17/12], [3/2, 5/6, 7/6], 27648*x^2) is conjectured to be algebraic over Q(x).

Conjectural: a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for  prime p >= 5 and positive integers n and k.

EXAMPLE

a(11) - a(1) = 187495217080545878999040 - 40 = (2^3)*(5^3)*(11^3)*140867931690868429 == 0 (mod 11^3).

MAPLE

seq(binomial(6*n, 2*n)*binomial(4*n, n)/binomial(3*n/2, n), n = 0..13);

CROSSREFS

Cf. A295431, A347855 - A347858.

Sequence in context: A059948 A229604 A045502 * A259461 A229692 A270053

Adjacent sequences:  A347851 A347852 A347853 * A347855 A347856 A347857

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Sep 16 2021

STATUS

approved

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Last modified January 24 18:21 EST 2022. Contains 350565 sequences. (Running on oeis4.)