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

1,2

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.

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

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 *)

Crossrefs

Sequence in context: A103971 A270085 A035406 * A034190 A216390 A305476

Adjacent sequences: A103929 A103930 A103931 * A103933 A103934 A103935

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Mar 24 2005

Status

approved