A364181 a(n) = (10*n)!*(3*n/2)!/((5*n)!*(9*n/2)!*(2*n)!).

1, 384, 461890, 638582784, 935387159850, 1414457284624384, 2182519096151533552, 3414991108739243704320, 5398397695681095146608490, 8600772808890306913527398400, 13787702861800799166026014363140, 22213518902232966637201617101783040, 35936545440404705429404600374145350960

Offset: 0

Author

Peter Bala, Jul 13 2023

Keywords

• nonn
• easy

Comments

A295475, defined by A295475(n) = (20*n)!*(3*n)! / ((10*n)!*(9*n)!*(4*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 45). Here we are essentially considering the sequence {A295475(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, A295475, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

Programs

Maple

seq( simplify((10*n)!*(3*n/2)!/((5*n)!*(9*n/2)!*(2*n)!)), n = 0..15);

Formula

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (2^11)*(5^5)/(3^8)*sqrt(3).

a(n) = 409600*(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)*(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) = 384