A105639 - OEIS

%I #12 Aug 11 2017 06:09:32

%S 0,1,3,51,2635,321315,79244571,35534634163,26790753983211,

%T 31980883597248195,57639013468037578555,150903079070698932214611,

%U 555841597474333410204232203,2804056152239296833617706906211,18933384891214439885244043983467355

%N Multiples of coefficients in an asymptotic series of Ramanujan.

%D G. H. Hardy, Srinivasa Ramanujan (1887-1920), pp. xxi-xxxvi of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page xxvi VII. (4)

%H Seiichi Manyama, <a href="/A105639/b105639.txt">Table of n, a(n) for n = 0..158</a>

%F a(n) = -G(n)G(n+1) where G=A001469 Genocchi numbers.

%F Sum_{k>0} k^2/(e^(k*x) - 1) = zeta(3)*2/x^3 - 1/(12*x) + Sum_{k>0} a(k)x^(2*k-1)/((2*k)!(2*k+2)*4*(2^(2*k)-1)*(2^(2*k+2)-1)).

%e x/1440 + x^3/181440 + x^5/7257600 + x^7/159667200 + 691x^9/1569209241600 + ...

%o (PARI) a(n)=if(n<0,0, n*=2; -4*(2^n-1)*(4*2^n-1)*bernfrac(n)*bernfrac(n+2))

%Y Cf. A001469.

%K nonn

%O 0,3

%A _Michael Somos_, Apr 16 2005