A103932 Numerators of first difference of squares of harmonic numbers.

1, 5, 10, 47, 131, 71, 353, 1487, 6989, 1451, 82451, 42433, 1132133, 1158863, 236749, 4828073, 41781863, 42482563, 273253759, 277235737, 56204647, 18975625, 441730115, 670193263, 33874048171, 34224132367, 311048966203, 313970420453

Offset: 1

Author

Wolfdieter Lang, Mar 24 2005

Keywords

• nonn
• easy
• frac

Comments

The corresponding denominators are given in A103933.

h(n+1) + h(n) = (n+1)*(h(n+1)^2 - h(n)^2), where h(n) is the n-th harmonic number. - Gary Detlefs, May 25 2012

Links

• Robert Israel, Table of n, a(n) for n = 1..2296
• Wolfdieter Lang, Rationals.

Programs

Maple

H:= Vector(51):
for i from 2 to 51 do H[i]:= H[i-1]+1/(i-1) od:
HS:= map(t -> t^2, H):
convert(map(numer, HS[2..-1]-HS[1..-2]),list); # Robert Israel, Sep 27 2023

Mathematica

Array[ HarmonicNumber[#]^2&, 29, 0] // Differences // Numerator (* Jean-François Alcover, Jul 09 2013 *)

Formula

a(n) = numerator(r(n)), with the rationals r(n) = H(n)^2 - H(n-1)^2 where H(n) = A001008(n)/A002805(n), n >= 1, H(0):=0.

G.f. for r(n): (log(1-x))^2 + dilog(1-x) where dilog(1-x) = polylog(2, x).

a(n) = numerator(h(n) + h(n-1)), where h(n) is the n-th harmonic number. - Gary Detlefs, May 25 2012