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

1,2

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

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).

Example

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

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

Prog

(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

Crossrefs

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

Sequence in context: A286780 A099885 A106372 * A214916 A323291 A161203

Adjacent sequences: A354892 A354893 A354894 * A354896 A354897 A354898

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Jun 10 2022

Status

approved