A105639 Multiples of coefficients in an asymptotic series of Ramanujan.

0, 1, 3, 51, 2635, 321315, 79244571, 35534634163, 26790753983211, 31980883597248195, 57639013468037578555, 150903079070698932214611, 555841597474333410204232203, 2804056152239296833617706906211, 18933384891214439885244043983467355

Offset

0,3

References

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)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..158

Formula

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

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)).

Example

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

Prog

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

Crossrefs

Cf. A001469.

Sequence in context: A377493 A172434 A210676 * A003028 A069343 A084882

Adjacent sequences: A105636 A105637 A105638 * A105640 A105641 A105642

Keyword

nonn

Author

Michael Somos, Apr 16 2005

Status

approved