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%I #16 Jan 01 2022 16:09:03
%S 1,8,184,8448,648576,74972160,12174658560,2643856588800,
%T 740051782041600,259500083163955200,111422936937037824000,
%U 57504006817918746624000,35122852492484487413760000
%N Scaled sums of odd reciprocals.
%C a(n) mod n^2 = 2*n if n is an odd prime, otherwise 0. - _Gary Detlefs_, Apr 16 2012
%H Harvey P. Dale, <a href="/A049034/b049034.txt">Table of n, a(n) for n = 0..224</a>
%F a(n) = (2*n+1)! * sum[ k=0..n ] 1/(2*k+1).
%F 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)!.
%e (arctanh x)^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...
%t Module[{nn=25,c},c=Range[1,nn,2];Times@@@Thread[{Accumulate[1/c],c!}]](* _Harvey P. Dale_, Nov 20 2013 *)
%o (PARI) {a(n)=if(n<0, 0, n=2*n+1; n!*sum(k=1, n, (k%2)/k))} /* _Michael Somos_, Sep 19 2006 */
%Y Bisection of A081358 and A092692: a(n) = A081358(2n+1) = A092692(2n+1).
%Y Cf. A000254, A004041, A046674.
%K nonn
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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