OFFSET
1,2
COMMENTS
Numerator of a second-order harmonic number H(n, (2)) = Sum_{k=1..n} HarmonicNumber(k). - Alexander Adamchuk, Apr 12 2006
p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 06 2006
Denominator is A027611(n+1). p divides a(p-3) for prime p > 3. - Alexander Adamchuk, Jul 26 2006
a(n) = A213998(n,n-2) for n > 1. - Reinhard Zumkeller, Jul 03 2012
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number.
FORMULA
From Vladeta Jovovic, Sep 02 2002: (Start)
a(n) = numerators of coefficients in expansion of -log(1-x)/(1-x)^2.
a(n) = numerators of (n+1)*(harmonic(n+1) - 1).
a(n) = numerators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)
a(n) = numerator( Sum_{k=1..n} Sum_{i=1..k} 1/i ). - Alexander Adamchuk, Apr 12 2006
a(n) = numerator( Sum_{k=1..n} k/(n-k+1) ). - Alexander Adamchuk, Jul 26 2006
a(n) = numerator of integral_{x=1..n+1} floor((n+1)/x). - Jean-François Alcover, Jun 18 2013
MAPLE
a := n -> numer(add((n+1-j)/j, j=1..n));
seq(a(n), n = 1..29); # Peter Luschny, May 12 2023
MATHEMATICA
Numerator[Table[Sum[Sum[1/i, {i, 1, k}], {k, 1, n}], {n, 1, 30}]] (* Alexander Adamchuk, Apr 12 2006 *)
Numerator[Table[Sum[k/(n-k+1), {k, 1, n}], {n, 1, 50}]] (* Alexander Adamchuk, Jul 26 2006 *)
PROG
(Haskell)
import Data.Ratio ((%), numerator)
a027612 n = numerator $ sum $ zipWith (%) [1 .. n] [n, n-1 .. 1]
-- Reinhard Zumkeller, Jul 03 2012
(PARI) a(n) = numerator(sum(k=1, n, k/(n-k+1))); \\ Michel Marcus, Jul 14 2018
(Magma) [Numerator((&+[j/(n-j+1): j in [1..n]])): n in [1..30]]; // G. C. Greubel, Aug 23 2022
(SageMath) [numerator(n*(harmonic_number(n+1) - 1)) for n in (1..30)] # G. C. Greubel, Aug 23 2022
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Glen Burch (gburch(AT)erols.com)
STATUS
approved