A354895 a(n) is the denominator of the n-th hyperharmonic number of order n.

1, 2, 6, 12, 20, 60, 210, 56, 504, 504, 660, 3960, 5148, 4004, 4290, 34320, 17680, 31824, 302328, 77520, 813960, 8953560, 2288132, 27457584, 49031400, 12498200, 168725700, 42948360, 10925460, 163881900, 2540169450, 645122400, 327523680, 5567902560, 1412149200

Offset: 1

Author

Ilya Gutkovskiy, Jun 10 2022

Keywords

• nonn
• frac

Examples

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...
		

References

• J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.

Links

• Robert Israel, Table of n, a(n) for n = 1..3764
• Eric Weisstein's World of Mathematics, Harmonic Number
• Wikipedia, Hyperharmonic number

Crossrefs

Cf. A001700, A002805, A027611, A058806, A076175, A124838, A354894 (numerators).

Programs

Maple

N:= 100: # for a(1)..a(N)
H:= ListTools:-PartialSums([seq(1/i,i=1..2*N-1)]):
f:= n -> denom(binomial(2*n-1,n-1)*(H[2*n-1]-H[n-1])):
f(1):= 1:
map(f, [$1..N]); # Robert Israel, Jul 10 2023

Mathematica

Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 35}] // Denominator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 35}] // Denominator

PARI

H(n) = sum(i=1, n, 1/i);
a(n) = denominator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022

Python

from math import comb
from sympy import harmonic
def A354895(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).q # Chai Wah Wu, Jun 18 2022

Formula

a(n) is the denominator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.

a(n) is the denominator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.

A354894(n) / a(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).