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A354894
a(n) is the numerator of the n-th hyperharmonic number of order n.
2
1, 5, 47, 319, 1879, 20417, 263111, 261395, 8842385, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773, 154479849447926113, 6637417807457499259, 6632660439700528339
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
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
Differs from A049281.
Sequence in context: A344121 A304371 A049281 * A219073 A327595 A353545
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, Jun 10 2022
STATUS
approved