A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.

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

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

• Eric Weisstein's World of Mathematics, Harmonic Number
• Wikipedia, Hyperharmonic number

Crossrefs

Cf. A001008, A001700, A027612, A058806, A076174, A124837, A354895 (denominators).

Differs from A049281.

Programs

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

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

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).