A145623 Numerator of the polynomial A_l(x) = Sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=8.

68, 13126, 4200532, 1881839401, 361313167484, 254364469931206, 211631238983010892, 5417759717965164721, 2947261286573050252868, 17919348622364145592266214, 1146838311831305317954669876

Offset: 1

Author

Artur Jasinski, Oct 14 2008

Keywords

• frac
• nonn

Comments

For denominators see A145624. For general properties of A_l(x) see A145609.

Links

• Robert Israel, Table of n, a(n) for n = 1..375

Crossrefs

Cf. A145609 - A145640.

Programs

Maple

G:= (32*sqrt(x)*ln((1-sqrt(x))/(1+sqrt(x))) + 4*ln(1-x))/(64*x-1):
S:= series(G, x, 51):
seq(coeff(S,x,n),n=1..50); # Robert Israel, Mar 09 2016

Mathematica

m = 8; 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, Oct 14 2008 *)
a[n_,m_]:=Integrate[(m-x^n)/(m-x),{x,0,1}]+(m^n-m)Log[m/(m-1)]
Table[8 a[2 n, 8] // Simplify  // Numerator, {n,1,25}]  (* Gerry Martens , Jun 04 2016 *)

Formula

Sum_{n >= 1} (a(n)/A145624(n))*x^n = (32*sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) - 4*log(1-x))/(1-64*x). - Robert Israel, Mar 09 2016

Extensions

Edited by R. J. Mathar, Aug 21 2009