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
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, Jun 10 2022
STATUS
approved