OFFSET
1,2
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.
LINKS
Eric Weisstein's World of Mathematics, Harmonic Number
Wikipedia, Hyperharmonic number
FORMULA
a(n) is the numerator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
a(n) is the numerator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.
a(n) / A354895(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, ...
MATHEMATICA
Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 23}] // Numerator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 23}] // Numerator
PROG
(PARI) H(n) = sum(i=1, n, 1/i);
a(n) = numerator(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 A354894(n): return (comb(2*n-1, n-1)*(harmonic(2*n-1)-harmonic(n-1))).p # Chai Wah Wu, Jun 18 2022
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, Jun 10 2022
STATUS
approved