A364183 a(n) = (12*n)!*(2*n)!*(n/2)!/((6*n)!*(4*n)!*(7*n/2)!*n!).

1, 4224, 76488984, 1626105446400, 36856530424884600, 864687003650148532224, 20728451893251973782071160, 504292670666772382512278667264, 12401082728528113445556802226795640, 307453669544695584297743425538327838720, 7671567513095586883562392061857092727662984

Offset: 0

Author

Peter Bala, Jul 13 2023

Keywords

• nonn
• easy

Comments

A295479, defined by A295479(n) = (24*n)!*(4*n)!*n! / ((12*n)!*(8*n)!*(7*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 49). Here we are essentially considering the sequence {A295479(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

• 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, A295479, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

Programs

Maple

seq( simplify((12*n)!*(2*n)!*(n/2)!/((6*n)!*(4*n)!*(7*n/2)!*n!)), n = 0..15);

Formula

a(n) ~ c^n * 1/sqrt(14*Pi*n), where c = (2^15)*(3^6)/(7^4) * sqrt(7).

a(n) = 1327104*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(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) = 4224.