A145609 Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=1.

3, 25, 49, 761, 7381, 86021, 1171733, 2436559, 14274301, 55835135, 19093197, 1347822955, 34395742267, 315404588903, 9304682830147, 586061125622639, 54062195834749, 54801925434709, 2053580969474233, 2078178381193813

Offset: 1

Author

Artur Jasinski, Oct 14 2008

Keywords

• frac
• nonn

Comments

The polynomials A_{2n+1}(x) = sum_{d=1..2n} x^(2n+1-d)/d for small n look as follows:

n=1, index = 3: A_3(x) = x/2 + x^2.

n=2, index = 5: A_5(x) = x/4 + x^2/3 + x^3/2 + x^4.

n=3, index = 7: A_7(x) = x/6 + x^2/5 + x^3/4 + x^4/3 + x^5/2 + x^6.

n=4, index = 9: A_9(x) = x/8 + x^2/7 + x^3/6 + x^4/5 + x^5/4 + x^6/3 + x^7/2 + x^8.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

For denominators see A145610.

Cf. A145611-A145640.

Programs

Maple

A := proc(l,x) add(x^(l-d)/d,d=1..l-1) ; end: A145609 := proc(n) numer( A(2*n+1,1)) ; end: seq(A145609(n),n=1..20) ; # R. J. Mathar, Aug 21 2009

Mathematica

m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (* Artur Jasinski *)

Formula

(1/(2*n+1))*2F1(1, 2*n+1; 2*n+2; 1/m) = Sum_{x>=0} m^(-x)/(x+2n+1) = m^(2n)*arctanh((2m-1)/(2m^2-2m+1)) - A_{2n+1}(m) = m^(2n)*log(m/(m-1)) - A_{2n+1}(m). - Artur Jasinski, Oct 14 2008

It appears that A145609(n)/A145610(n) = H(2*n+2), the harmonic number of order 2*n+2. - Groux Roland, Jan 08 2011

Yes, A145609(n)/A145610(n) = H(2*n+2), as A_l(x) = sum_{d=1..l-1} x^(l-d)/d at x=1 is just sum_{d=1..l-1}1/d = H(l-1), the harmonic number of order (l-1). - Zak Seidov, Jan 09 2014

a(n) = numerator of Integral_{x=0..1} (1 - x^(2*n))/(1 - x). - Peter Luschny, Sep 28 2017

Extensions

Edited, parentheses in front of Gauss. Hypg. Fct. added by R. J. Mathar, Aug 21 2009