A007779 Coefficients of asymptotic expansion of Ramanujan false theta series.

1, 1, 1, 2, 5, 17, 72, 367, 2179, 14750, 112023, 942879, 8708912, 87563937, 951933849, 11125383714, 139092236301, 1852257089937, 26173848663000, 391153031777263, 6163682285356171, 102136840106457790, 1775499429402739247, 32307194057014483391

Offset: 0

Author

William F. Galway (galway(AT)math.uiuc.edu)

Keywords

• nonn
• nice
• easy

Comments

Also a(n) = number of alternating fixed-point-free involutions on 1,2,...,2n, i.e., w(1) > w(2) < w(3) > w(4) < ... > w(2n), w^2=1 and w(i) != i for all i. - Richard Stanley, Jan 22 2006. For example, a(3)=2 because there are two alternating fixed-point-free involutions on 1,...,6, viz., 214365 and 645231.

If b(n) is the number of reverse alternating fixed-point-free involutions on 1,2,...,2n (A115455) then b(n-1) + b(n) = a(n). - Richard Stanley, Jan 22 2006

References

• B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 545.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• W. F. Galway, An Asymptotic Expansion of Ramanujan, in Number Theory (Fifth Conference of Canadian Number Theory Assoc., August, 1996, Carleton University), pp. 107-110, ed. R. Gupta and K. S. Williams, Amer. Math. Soc., 1999.
• R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006.
• R. P. Stanley, Permutations

Crossrefs

Cf. A000111, A115455.

Programs

Mathematica

Table[SeriesCoefficient[(1-x^2)^(-1/4)*(1+x)^(1/2)*Sum[(-1)^k*EulerE[2*k]*(1/4*Log[(1+x)/(1-x)])^k/k!,{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 29 2014 *)

Formula

Sum_{n>=0} a(n)*x^n = (1-x^2)^(-1/4)*sqrt(1+x)*Sum_{k>=0} E_{2k} v^k/k!, where E_{2k} is an Euler number and v = (1/4)*log((1+x)/(1-x)). - Richard Stanley, Jan 22 2006

Berndt gives an explicit g.f. on page 547.

Extensions

Edited by Ralf Stephan, May 08 2007