cp's OEIS Frontend

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

%I A364180 #13 Jul 16 2023 05:49:06
%S A364180 1,1152,5542680,31473008640,190818980609400,1198265754978353152,
%T A364180 7691041400616850556280,50107639155283424528302080,
%U A364180 330014847932376708502470210680,2191489080600524699617120065945600,14647137653300940580784413641872332680
%N A364180 a(n) = (10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!).
%C A364180 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).
%C A364180 This sequence is only conjecturally an integer sequence.
%C A364180 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.
%H A364180 J. W. Bober, <a href="http://arxiv.org/abs/0709.1977">Factorial ratios, hypergeometric series, and a family of step functions</a>, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
%F A364180 a(n) ~ c^n * 1/sqrt(14*Pi*n), where c = (2^11)*(5^5)/(7^4) * sqrt(7).
%F A364180 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.
%p A364180 seq( simplify((10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!)), n = 0..15);
%Y A364180 Cf. A061164, A276100, A276101, A276102, A295431, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.
%K A364180 nonn,easy
%O A364180 0,2
%A A364180 _Peter Bala_, Jul 13 2023