A364184 a(n) = (12*n)!*(2*n)!*(3*n/2)!/((6*n)!*(9*n/2)!*(4*n)!*n!).

1, 1408, 6374082, 32993443840, 180669266788650, 1020694137466257408, 5882199787281395215344, 34369110490167819009785856, 202857467914154836183288657770, 1206640354461153104738279049134080, 7221430962039777689508936047385667332

Offset: 0

Author

Peter Bala, Jul 13 2023

Keywords

• nonn
• easy
• changed

Comments

A295481, defined by A295481(n) = (24*n)!*(4*n)!*(3*n)! / ((12*n)!*(9*n)!*(8*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 51). Here we are essentially considering the sequence {A295481(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (9*n/2)! := Gamma(1 + 9*n/2).

This sequence is only conjecturally an integer sequence.

Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

Links

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

Crossrefs

Cf. A276100, A276101, A276102, A295431, A295481, A347854, A347855, A347856, A347857, A347858, A364173 - A364183.

Programs

Maple

seq( simplify((12*n)!*(2*n)!*(3*n/2)!/((6*n)!*(9*n/2)!*(4*n)!*n!)), n = 0..15);

Formula

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (2^15)/(3^2) * sqrt(3).

a(n) = 49152*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(n*(n - 1)*(9*n - 2)*(9*n - 4)*(9*n - 8)*(9*n - 10)*(9*n - 14)*(9*n - 16))*a(n-2) with a(0) = 1 and a(1) = 1408.