A211418 -id:A211418 - cp's OEIS frontend

A276100 a(n) = (15n)!(n/2)!/((15n/2)!(5n)!(3*n)!).

1, 114688, 77636318760, 62505037015810048, 53837289804317953893960, 48066503353826060675410034688, 43880754270176401422739454033276880, 40671547154451909281150562260837340282880, 38113558705192522309151157825210540422513019720

Offset: 0

Peter Bala, Aug 22 2016

Fractional factorials are defined in terms of the gamma function, for example, (n/2)! := gamma(n/2 + 1).
This is only conjecturally an integer sequence. The similarly defined sequence (15*n)!*floor(n/2)!/(floor(15*n/2)!*(5*n)!*(3*n)!) = A211418(15*n) is integral.
Let u(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!) = A211417(n). This sequence of ratios of factorials is integral and was used by Chebyshev in his estimate of the number of primes less than or equal to a fixed integer n. The three sequences u(1/2*n), u(1/3*n) and u(1/5*n) also appear to be integral (checked up to n = 200). This is the sequence u(1/2*n). See A276101( u(1/3*n) ) and A276102( u(1/5*n) ).
The generating function for u(n) is Hypergeom([29/30, 23/30, 19/30, 17/30, 13/30, 11/30, 7/30, 1/30], [4/5, 3/5, 2/5, 1/5, 2/3, 1/3, 1/2], (2^14*3^9*5^5)*x) and is known to be algebraic - see Rodriguez-Villegas. Are the generating functions for u(1/2*n), u(1/3*n) and u(1/5*n) also algebraic?

Carauleanu Marc, Table of n, a(n) for n = 0..111
P. Bala, Some integer ratios of factorials
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Cf. A211417, A211418, A276101, A276102.

A211417 := proc(n)
(30*n)!*(n)!/((15*n)!(10*n)!(6*n)!);
end proc:
seq(simplify(A211417(1/2*n)), n = 0..10);

Table[(15 n)!*(n/2)!/((15 n/2)!*(5 n)!*(3 n)!), {n, 0, 8}] (* Michael De Vlieger, Aug 28 2016 *)

O.g.f.: A(x) = Hypergeom([29/30, 23/30, 19/30, 17/30, 13/30, 11/30, 7/30, 1/30,], [4/5, 3/5, 2/5, 1/5, 2/3, 1/3, 1/2], (2^14*3^9*5^5)*x^2) + 114688*x*Hypergeom([22/15, 19/15, 17/15, 16/15, 14/15, 13/15, 11/15, 8/15], [13/10, 11/10, 9/10, 7/10, 7/6, 5/6, 3/2], (2^14*3^9*5^5)*x^2).

a(n) ~ (2^14*3^9*5^5)^(n/2)/sqrt(30*Pi*n).

A276101 a(n) = (10n)!(n/3)!/((5n)!(10n/3)!(2*n)!).

Original entry on oeis.org

1, 1458, 9723402, 77636318760, 665145965903562, 5915482311008318958, 53837289804317953893960, 497704257299202369371725086, 4653371135224869009103021872330, 43880754270176401422739454033276880

Offset: 0

Peter Bala, Aug 22 2016

Fractional factorials are defined in terms of the gamma function, for example, (n/3)! := gamma(n/3 + 1).
This is only conjecturally an integer sequence. The similarly defined sequence (10*n)!*floor(n/3)!/((5*n)!*floor(10*n/3)!*(2*n)!) = A211418(10*n) is integral.
Let u(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!) = A211417(n). The three sequences u(1/2*n), u(1/3*n) and u(1/5*n) appear to be integral (checked up to n = 200). This is the sequence u(1/3*n). See A276100 ( u(1/2*n) ) and A276102 ( u(1/5*n) ).
The generating function for u(n) is Hypergeom([29/30, 23/30, 19/30, 17/30, 13/30, 11/30, 7/30, 1/30], [4/5, 3/5, 2/5, 1/5, 2/3, 1/3, 1/2], (2^14*3^9*5^5)*x) and is known to be algebraic. Are the generating functions for u(1/2*n), u(1/3*n) and u(1/5*n) also algebraic?

P. Bala, Some integer ratios of factorials
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Cf. A211417, A276100, A276102.

A211417 := proc(n)
(30*n)!*(n)!/((15*n)!(10*n)!(6*n)!);
end proc:
seq(simplify(A211417(1/3*n)), n = 0..10);

Table[(10*n)!*(n/3)!/((5*n)!*(10*n/3)!*(2*n)!) // FullSimplify, {n, 0, 9}] (* Jean-François Alcover, Nov 27 2017 *)

O.g.f. A(x) = Hypergeom([29/30, 23/30, 19/30, 17/30, 13/30, 11/30, 7/30, 1/30], [4/5, 3/5, 2/5, 1/5, 2/3, 1/3, 1/2], (2^14*3^9*5^5)*x^3) + 1458*x*Hypergeom([29/30, 23/30, 17/30, 11/30, 13/10, 11/10, 9/10, 7/10], [17/15, 14/15, 11/15, 8/15, 5/6, 4/3, 2/3], (2^14*3^9*5^5)*x^3) + 9723402*x^2*Hypergeom([ 49/30, 43/30, 37/30, 31/30, 13/10, 11/10, 9/10, 7/10], [22/15, 19/15, 16/15, 13/15, 7/6, 5/3, 4/3],(2^14*3^9*5^5)*x^3).

a(n) ~ (2^14*3^9*5^5)^(n/3)/(sqrt(20*Pi*n)).

A276102 a(n) = (6n)!(n/5)!/((3n)!(2n)!(6*n/5)!).

Original entry on oeis.org

1, 50, 8250, 1636250, 349456250, 77636318760, 17672894531250, 4089765214843750, 957711284472656250, 226280605806640625000, 53837289804317953893960, 12880759628253295898437500

Offset: 0

Peter Bala, Aug 22 2016

Fractional factorials are defined in terms of the gamma function, for example, (n/5)! := gamma(n/5 + 1).
This is only conjecturally an integer sequence. The similarly defined sequence (6*n)!*floor(n/5)!/((3*n)!*(2*n)!*floor(6*n/5)!) = A211418(6*n) is integral.
Let u(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!) = A211417(n). The three sequences u(1/2*n), u(1/3*n) and u(1/5*n) appear to be integral (checked up to n = 200). This is the sequence u(1/5*n). See A276100 ( u(1/2*n) ) and A276101 ( u(1/3*n) ).
The generating function for u(n) is Hypergeom([29/30, 23/30, 19/30, 17/30, 13/30, 11/30, 7/30, 1/30], [4/5, 3/5, 2/5, 1/5, 2/3, 1/3, 1/2], (2^14*3^9*5^5)*x) and is algebraic - see Rodriguez-Villegas. Are the generating functions for u(1/2*n), u(1/3*n) and u(1/5*n) also algebraic?
The o.g.f. for this sequence can be expressed as a sum of 5 generalized hypergeometric functions of type 8F7.

P. Bala, Some integer ratios of factorials
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Cf. A211417, A276100, A276101.

A211417 := proc(n)
(30*n)!*(n)!/((15*n)!(10*n)!(6*n)!);
end proc:
seq(simplify(A211417(1/5*n)), n = 0..10);

Table[(6*n)!*(n/5)!/((3*n)!*(2*n)!*(6*n/5)!) // FullSimplify, {n, 0, 11}] (* Jean-François Alcover, Nov 27 2017 *)

a(n) ~ (2^14*3^9*5^5)^(n/5)/sqrt(12*Pi*n).

A211417 Integral factorial ratio sequence: a(n) = (30n)!n!/((15n)!(10n)!(6*n)!).

Original entry on oeis.org

1, 77636318760, 53837289804317953893960, 43880754270176401422739454033276880, 38113558705192522309151157825210540422513019720, 34255316578084325260482016910137568877961925210286281393760

Offset: 0

Peter Bala, Apr 11 2012

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.
This sequence is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin.
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.
See the Bala link for the proof that a(n), n = 0,1,2..., is an integer.
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

N. J. A. Sloane, Table of n, a(n) for n = 0..50
Peter Bala, Proof of the integrality of A211417 and A211418
Frits Beukers, Hypergeometric functions, how special are they?, Notices Amer. Math. Soc. 61 (2014), no. 1, 48--56. MR3137256
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.
Florian Fürnsinn and Sergey Yurkevich, Algebraicity of hypergeometric functions with arbitrary parameters, arXiv:2308.12855 [math.CA], 2023.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Fernando Rodriguez Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math.NT/0701362, 2007.
Fernando Rodriguez Villegas, Mixed Hodge numbers and factorial ratios, arXiv:1907.02722 [math.NT], 2019.
K. Soundararajan, Integral Factorial Ratios, arXiv:1901.05133 [math.NT], 2019.
Wadim Zudilin, Integer-valued factorial ratios, MathOverflow question 26336, 2010.

Cf. A182067, A211418, A061162, A061163, A061164, A091496, A091527, A112292, A182400, A211419, A211420, A211421, A276100, A262733, A295431.

[Factorial(30*n)*Factorial(n)/(Factorial(15*n)*Factorial(10*n)*Factorial(6*n)): n in [0..10]]; // Vincenzo Librandi, Oct 03 2015

Table[(30 n)!*n!/((15 n)!*(10 n)!*(6 n)!), {n, 0, 5}] (* Michael De Vlieger, Oct 02 2015 *)

a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!);
vector(10, n, a(n-1)) \\ Altug Alkan, Oct 02 2015

a(n) ~ 2^(14*n-1) * 3^(9*n-1/2) * 5^(5*n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 30 2016

cp's OEIS Frontend

A276100 a(n) = (15*n)!*(n/2)!/((15*n/2)!*(5*n)!*(3*n)!).

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A276101 a(n) = (10*n)!*(n/3)!/((5*n)!*(10*n/3)!*(2*n)!).

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A276102 a(n) = (6*n)!*(n/5)!/((3*n)!*(2*n)!*(6*n/5)!).

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A211417 Integral factorial ratio sequence: a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!).

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Magma

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Formula

A276100 a(n) = (15n)!(n/2)!/((15n/2)!(5n)!(3*n)!).

A276101 a(n) = (10n)!(n/3)!/((5n)!(10n/3)!(2*n)!).

A276102 a(n) = (6n)!(n/5)!/((3n)!(2n)!(6*n/5)!).

A211417 Integral factorial ratio sequence: a(n) = (30n)!n!/((15n)!(10n)!(6*n)!).