A076175 -id:A076175 - cp's OEIS frontend

A076174 Numerator of Sum_{i+j+k=n, i,j,k>=1} (i*j)/k.

0, 0, 1, 9, 37, 319, 743, 2509, 2761, 32891, 35201, 485333, 511073, 535097, 1115239, 19679783, 6786821, 133033679, 136913555, 140608675, 144135835, 678544345, 693417203, 17692378667, 18035598467, 165294957803, 168163294703

Offset: 1

Benoit Cloitre, Nov 01 2002

a(n) is odd.

a(n+2) = Numerators of 4th-order harmonic numbers (defined by Conway and Guy, 1996). - Alexander Adamchuk, Jun 14 2008

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.

Alexander Adamchuk, Jun 14 2008, Table of n, a(n) for n = 1..52

Cf. A076175.

Cf. A124837 = Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Table[ Numerator[Sum[ Sum[ Sum[ Sum[ 1/k, {k,1,l} ], {l,1,m} ], {m,1,n} ], {n,1,s-2} ] ], {s,1,52} ] Table[ Numerator[ (n-1)n(n+1)/6 * Sum[ 1/k, {k,4,n+1} ] ], {n,1,50}] (* Alexander Adamchuk, Jun 14 2008 *)

a(n)=numerator(sum(i=1,n,sum(j=1,n,sum(k=1,n,if(n-i-j-k,0,1)*i*j/k))))

a(n) = Numerator[Sum[ Sum[ Sum[ Sum[ 1/k, {k,1,l} ], {l,1,m} ], {m,1,n} ], {n,1,s-2} ] ]. a(n) = Numerator[ (n-1)n(n+1)/6 * Sum[ 1/k, {k,4,n+1} ] ]. - Alexander Adamchuk, Jun 14 2008

a(n) = Numerator(sum(1/(k+3), k=1..n-2)), n>1. - Gary Detlefs, Sep 14 2011

A354895 a(n) is the denominator of the n-th hyperharmonic number of order n.

Original entry on oeis.org

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

Ilya Gutkovskiy, Jun 10 2022

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.

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

Cf. A001700, A002805, A027611, A058806, A076175, A124838, A354894 (numerators).

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

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

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

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

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).

cp's OEIS Frontend

A076174 Numerator of Sum_{i+j+k=n, i,j,k>=1} (i*j)/k.

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