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A364182 a(n) = (12*n)!*(n/2)!/((6*n)!*(4*n)!*(5*n/2)!). 0
1, 7392, 267711444, 11489451294720, 527048385075849780, 25051434899696246587392, 1217325447549161369383451760, 60050961586064738516089033457664, 2994861478939539397101967737771147060, 150602318360773064327512837557840362078208 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A295477, defined by A295477(n) = (24*n)!*n! / ((12*n)!*(8*n)!*(5*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 47). Here we are essentially considering the sequence {A295477(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (5*n/2)! := Gamma(1 + 5*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.
FORMULA
a(n) ~ c^n * 1/sqrt(20*Pi*n), where c = (2^12)*(3^6)/(5^3) * sqrt(5).
a(n) = 82944*(12*n - 1)*(12*n - 5)(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(5*n*(n - 1)*(2*n - 1)*(2*n - 3)*(5*n - 2)*(5*n - 4)*(5*n - 6)*(5*n - 8))*a(n-2) with a(0) = 1 and a(1) = 7392
MAPLE
seq( simplify((12*n)!*(n/2)!/((6*n)!*(4*n)!*(5*n/2)!)), n = 0..15);
CROSSREFS
Sequence in context: A028540 A347164 A206202 * A034279 A031584 A237311
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 13 2023
STATUS
approved

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Last modified August 25 06:02 EDT 2024. Contains 375422 sequences. (Running on oeis4.)