A364178 a(n) = (10*n)!*(3*n)!*(n/2)!/((6*n)!*(5*n)!*(3*n/2)!*n!).

1, 168, 83980, 48664320, 29966636700, 19075222663168, 12398706131799988, 8175717823943147520, 5447952226877283703580, 3659442300478634742251520, 2473617870747229982625186480, 1680586987551894402985233481728, 1146602219745194113307246953503300

Offset: 0

Author

Peter Bala, Jul 13 2023

Keywords

• nonn
• easy

Comments

A295470, defined by A295470(n) = (20*n)!*(6*n)!*n! / ((12*n)!*(10*n)!*(3*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 40). Here we are essentially considering the sequence {A295470(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, A295470, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

Programs

Maple

seq( simplify((10*n)!*(3*n)!*(n/2)!/((6*n)!*(5*n)!*(3*n/2)!*n!)), n = 0..15);

Formula

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (10/3)^5 * 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)/(27*n*(n - 1)*(3*n - 2)*(3*n - 4)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11))*a(n-2) with a(0) = 1 and a(1) = 168.