bibkey: bala2021fractionalfactorial
authors: Peter Bala
year: 2021
title: "OEIS A347854–A347858: fractional factorial ratios"
doi: null
url: https://oeis.org/A347854
claim: "The sequence is conjectured to be integral."
strata_touched:
  - D5/S3/Arith/FactorialRatio/OeisA347854BalaHalfIndexIntegral
  - D5/S3/Arith/FactorialRatio/OeisA347855BalaThirdIndexIntegral
  - D5/S3/Arith/FactorialRatio/OeisA347855BalaAuxBIntegral
  - D5/S3/Arith/FactorialRatio/OeisA347855BalaAuxCIntegral
  - D5/S3/Arith/FactorialRatio/OeisA347856BalaHalfIndexIntegral
  - D5/S3/Arith/FactorialRatio/OeisA347857BalaHalfIndexIntegral
  - D5/S3/Arith/FactorialRatio/OeisA347858BalaHalfIndexIntegral
license: citation-only
triage: anchor

Fractional factorial ratios

Verified locator

Source: https://oeis.org/A347854

Related source entries: https://oeis.org/A347855, https://oeis.org/A347856, https://oeis.org/A347857, https://oeis.org/A347858. Peter Bala, Some integer ratios of factorials, pp. 5–6: https://oeis.org/A276098/a276098.pdf. J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, Table 2, https://arxiv.org/abs/0709.1977v2, DOI 10.1112/jlms/jdn078. The floor functions used here are associated with its first eight sporadic ratios.

Source statements

A347854 (unpaginated)

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

Fractional factorials are defined using the Gamma function; for example, (n/2)! := Gamma(1 + n/2).

Here we are essentially considering the sequence ( u(n/2) )n>=0. The sequence is conjectured to be integral.

A347855 (unpaginated)

a(n) = (4n)!/((2n)!(n)!) * (n/3)!/(4n/3)!.

Fractional factorials are defined using the Gamma function; for example, (n/3)! := Gamma(1 + n/3). The sequence defined by u(n) = (12n)!n! / ((6n)!(4n)!(3*n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 1). See A295431. Here we are essentially considering the sequence (u(n/3))n>=0. The sequence is conjectured to be integral.

a(3n+1) = 3^(3n+1) * b(n), where b(n) = (12n+4)!/( (6n+2)!(3n+1)!Product_{k = 0..3n} (3n+3k+4) ) appears to be integral.

a(3n+2) = 3^(3n+2) * c(n), where c(n) = (12n+8)!/( (6n+4)!(3n+1)!Product_{k = 0..3n+2} (3n+3k+2) ) appears to be integral. (End)

A347856 (unpaginated)

a(n) = (6n)!/((4n)!n!) * (n/2)!/(3n/2)!.

Fractional factorials are defined using the Gamma function; for example, (n/2)! := Gamma(1 + n/2).

The sequence defined by u(n) = (12n)!/((8n)!(2n)!) * n!/(3*n)! is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 3). See A295433. Here we are essentially considering the sequence (u(n/2))n>=0. The sequence is conjectured to be integral.

A347857 (unpaginated)

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

Fractional factorials are defined using the Gamma function; for example, (3n/2)! := Gamma(1 + 3n/2). The sequence defined by u(n) = (12n)!(3n)!/((6n)!(4n)!(5n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 5). See A295435. Here we are essentially considering the sequence (u(n/2))n>=0. The sequence is conjectured to be integral.

A347858 (unpaginated)

a(n) = (9n)!/((3n)!(2n)!) * (n/2)!/(9*n/2)!.

Fractional factorials are defined using the Gamma function; for example, (9n/2)! := Gamma(1 + 9n/2).

The sequence defined by u(n) = (18n)!(n)!/((4n)!(6n)!(9*n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 7). See A295437. Here we are essentially considering the sequence (u(n/2))n>=0. The sequence is conjectured to be integral.