A364180 a(n) = (10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!).

1, 1152, 5542680, 31473008640, 190818980609400, 1198265754978353152, 7691041400616850556280, 50107639155283424528302080, 330014847932376708502470210680, 2191489080600524699617120065945600, 14647137653300940580784413641872332680

Offset

0,2

Comments

A061164, defined by A061164(n) = (20*n)!*n! / ((10*n)!*(7*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 43). Here we are essentially considering the sequence {A061164(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (7*n/2)! := Gamma(1 + 7*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

Table of n, a(n) for n=0..10.

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.

Formula

a(n) ~ c^n * 1/sqrt(14*Pi*n), where c = (2^11)*(5^5)/(7^4) * sqrt(7).

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)/(7*n*(n - 1)*(7*n - 2)*(7*n - 4)*(7*n - 6)*(7*n - 8)*(7*n - 10)*(7*n - 12))*a(n-2) with a(0) = 1 and a(1) = 1152.

Maple

seq( simplify((10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!)), n = 0..15);

Crossrefs

Cf. A061164, A276100, A276101, A276102, A295431, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

Sequence in context: A166490 A035761 A107557 * A330266 A020382 A185689

Adjacent sequences: A364177 A364178 A364179 * A364181 A364182 A364183

Keyword

nonn,easy

Author

Peter Bala, Jul 13 2023

Status

approved