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
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
- Robert Israel, Table of n, a(n) for n = 1..3764
- Eric Weisstein's World of Mathematics, Harmonic Number
- Wikipedia, Hyperharmonic number
Programs
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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
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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 -
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
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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
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).