A103930 Numerators of squares of harmonic numbers A001008/A002805.

1, 9, 121, 625, 18769, 2401, 131769, 579121, 50822641, 54479161, 7007531521, 7399612441, 1313299956049, 1372958223289, 1429834803049, 5936819760481, 1775966959381729, 203755669038601, 75787776947048401, 3117562300468225

Offset: 1

Author

Wolfdieter Lang, Mar 24 2005

Keywords

• nonn
• easy
• frac

Comments

The corresponding denominators are given in A103931.

Links

• W. Lang: Rationals H(n)^2.

Programs

Maple

A103930 := n->numer(sum(1/i, i=1..n)^2); seq(A103930(k), k=1..40); # Wesley Ivan Hurt, Sep 30 2013

Mathematica

a[n_] := HarmonicNumber[n]^2 // Numerator; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 16 2013 *)

Formula

G.f.: -((d^3/dx^3)((log(1-x))^3))/3 + dilog(1-x)/(1-x) = ((log(1-x)^2) + dilog(1-x))/(1-x) with dilog(1-x)=polylog(2, x).

First differences give A103932(n)/A103933(n).

a(n) = numerator(H(n)^2), with the harmonic numbers H(n) = A001008(n)/A002805(n), n >= 1.