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 A347855 a(n) = (4*n)!/((2*n)!*(n)!) * (n/3)!/(4*n/3)!. 1
 1, 9, 189, 4620, 120285, 3241134, 89237148, 2493521172, 70429218525, 2005604901300, 57481750139814, 1656023714623980, 47913489552349980, 1391243084942932620, 40519970408738302020, 1183237138556438547120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Fractional factorials are defined using the Gamma function; for example, (n/3)! := Gamma(1 + n/3). 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. Here we are essentially considering the sequence (u(n/3))n>=0. The sequence is conjectured to be integral. LINKS 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. FORMULA a(n) = binomial(4*n,2*n)*binomial(2*n,n)/binomial(4*n/3,n). a(3*n) = A295431. a(3*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(3*n-3); a(3*n+1) = 648*(12*n-1)*(12*n-7)*(16*n^2-1)/(n*(6*n-1)*(9*n^2-1))*a(3*n-2); a(3*n+2) = 648*(12*n+1)*(12*n+7)*(16*n^2-1)/(n*(3*n+1)*(3*n+2)*(6*n+1))*a(3*n-1); Asymptotics: a(n) ~ 1/(2*sqrt(Pi*n))*2^(10*n/3)*3^n as n -> infinity. O.g.f.: A(x) = hypergeom([11/12, 7/12, 5/12, 1/12], [2/3, 1/2, 1/3], 27648*x^3) + 9*x*hypergeom([11/12, 5/4, 5/12, 3/4], [5/6, 4/3, 2/3], 27648*x^3) + 189*x^2*hypergeom([19/12, 13/12, 5/4, 3/4], [7/6, 5/3, 4/3], 27648*x^3) is conjectured to be algebraic over Q(x). Conjectural congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. EXAMPLE Congruence: a(11) - a(1) = 1656023714623980 - 9 = (3^2)*7*(11^3)*17* 1161713471 == 0 (mod 11^3). MAPLE seq( (4*n)!/((2*n)!*(n)!) * GAMMA(1+n/3)/GAMMA(1+4*n/3), n = 0..15); CROSSREFS Cf. A259431, A347854 - A347858. Sequence in context: A124008 A033713 A067422 * A249932 A278751 A145240 Adjacent sequences:  A347852 A347853 A347854 * A347856 A347857 A347858 KEYWORD nonn,easy AUTHOR Peter Bala, Sep 17 2021 STATUS approved

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Last modified January 27 10:14 EST 2022. Contains 350607 sequences. (Running on oeis4.)