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%I #62 Aug 29 2023 11:51:43
%S 1,77636318760,53837289804317953893960,
%T 43880754270176401422739454033276880,
%U 38113558705192522309151157825210540422513019720,34255316578084325260482016910137568877961925210286281393760
%N Integral factorial ratio sequence: a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!).
%C The integrality of this sequence can be used to prove Chebyshev's estimate C(1)*x/log(x) <= #{primes <= x} <= C(2)*x/log(x), for x sufficiently large; the constant C(1) = 0.921292... and C(2) = 1.105550.... Chebyshev's approach used the related step function floor(x) -floor(x/2) -floor(x/3) -floor(x/5) +floor(x/30). See A182067.
%C This sequence is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin.
%C The o.g.f. sum {n >= 0} a(n)*z^n is a generalized hypergeometric series of type 8F7 (see Bober, Table 2, Entry 31) and is an algebraic function of degree 483840 over the field of rational functions Q(z) (see Rodriguez-Villegas). Bober remarks that the monodromy group of the differential equation satisfied by the o.g.f. is W(E_8), the Weyl group of the E_8 root system.
%C See the Bala link for the proof that a(n), n = 0,1,2..., is an integer.
%C Congruences: a(p^k) == a(p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integer k (write a(n) as C(30*n,15*n)*C(15*n,5*n)/C(6*n,n) and use equation 39 in Mestrovic, p. 12). More generally, the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) may hold for any prime p >= 5 and any positive integers n and k. Cf. A295431. - _Peter Bala_, Jan 24 2020
%H N. J. A. Sloane, <a href="/A211417/b211417.txt">Table of n, a(n) for n = 0..50</a>
%H Peter Bala, <a href="/A211417/a211417_1.txt">Proof of the integrality of A211417 and A211418</a>
%H Frits Beukers, <a href="http://www.ams.org/notices/201401/rnoti-p48.pdf">Hypergeometric functions, how special are they?</a>, Notices Amer. Math. Soc. 61 (2014), no. 1, 48--56. MR3137256
%H 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.
%H Florian Fürnsinn and Sergey Yurkevich, <a href="https://arxiv.org/abs/2308.12855">Algebraicity of hypergeometric functions with arbitrary parameters</a>, arXiv:2308.12855 [math.CA], 2023.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H Fernando Rodriguez Villegas, <a href="http://arxiv.org/abs/math/0701362">Integral ratios of factorials and algebraic hypergeometric functions</a>, arXiv:math.NT/0701362, 2007.
%H Fernando Rodriguez Villegas, <a href="https://arxiv.org/abs/1907.02722">Mixed Hodge numbers and factorial ratios</a>, arXiv:1907.02722 [math.NT], 2019.
%H K. Soundararajan, <a href="https://arxiv.org/abs/1901.05133">Integral Factorial Ratios</a>, arXiv:1901.05133 [math.NT], 2019.
%H Wadim Zudilin, <a href="http://mathoverflow.net/questions/26336/">Integer-valued factorial ratios</a>, MathOverflow question 26336, 2010.
%F a(n) ~ 2^(14*n-1) * 3^(9*n-1/2) * 5^(5*n-1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 30 2016
%t Table[(30 n)!*n!/((15 n)!*(10 n)!*(6 n)!), {n, 0, 5}] (* _Michael De Vlieger_, Oct 02 2015 *)
%o (PARI) a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!);
%o vector(10, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015
%o (Magma) [Factorial(30*n)*Factorial(n)/(Factorial(15*n)*Factorial(10*n)*Factorial(6*n)): n in [0..10]]; // _Vincenzo Librandi_, Oct 03 2015
%Y Cf. A182067, A211418, A061162, A061163, A061164, A091496, A091527, A112292, A182400, A211419, A211420, A211421, A276100, A262733, A295431.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Apr 11 2012
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