A049034 Scaled sums of odd reciprocals.

1, 8, 184, 8448, 648576, 74972160, 12174658560, 2643856588800, 740051782041600, 259500083163955200, 111422936937037824000, 57504006817918746624000, 35122852492484487413760000

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn

Comments

a(n) mod n^2 = 2*n if n is an odd prime, otherwise 0. - Gary Detlefs, Apr 16 2012

Examples

			(arctanh x)^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..224

Crossrefs

Bisection of A081358 and A092692: a(n) = A081358(2n+1) = A092692(2n+1).

Cf. A000254, A004041, A046674.

Programs

Mathematica

Module[{nn=25,c},c=Range[1,nn,2];Times@@@Thread[{Accumulate[1/c],c!}]](* Harvey P. Dale, Nov 20 2013 *)

PARI

{a(n)=if(n<0, 0, n=2*n+1; n!*sum(k=1, n, (k%2)/k))} /* Michael Somos, Sep 19 2006 */

Formula

a(n) = (2*n+1)! * sum[ k=0..n ] 1/(2*k+1).

E.g.f. (arctanh x)^2/2 = sum_n a(n)x^(2n+2)/(2n+2)! or (arctanh x)/(1-x^2) = sum_n a(n)x^(2n+1)/(2n+1)!.