A364179 a(n) = (10*n)!*(n/2)!/((5*n)!*(4*n)!*(3*n/2)!).

1, 840, 2771340, 10754814720, 44524428808860, 190847602744995840, 835982760936614190900, 3716634993696885851422720, 16702642470437308383606668060, 75679458912906782280286032887808, 345116202503279265243707597937393840, 1581997780375359530321517073184807976960

Offset: 0

Author

Peter Bala, Jul 13 2023

Keywords

• nonn
• easy

Comments

A295471, defined by A295471(n) = (20*n)!*n! / ((10*n)!*(8*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 41). Here we are essentially considering the sequence {A295471(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (3*n/2)! := Gamma(1 + 3*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, A295471, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

Programs

Maple

seq( simplify((10*n)!*(n/2)!/((5*n)!*(4*n)!*(3*n/2)!)), n = 0..15);

Formula

a(n) ~ c^n * 1 /sqrt(12*Pi*n), where c = (2^3)*(5^5)/(3^2) * sqrt(3).

a(n) = 1600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(n*(3*n - 1)*(3*n - 2)*(3*n - 4)*(4*n - 1)*(4*n - 3)*(4*n - 5)*(4*n - 7))*a(n-2) with a(0) = 1 and a(1) = 840.