login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007779 Coefficients of asymptotic expansion of Ramanujan false theta series. 4
1, 1, 1, 2, 5, 17, 72, 367, 2179, 14750, 112023, 942879, 8708912, 87563937, 951933849, 11125383714, 139092236301, 1852257089937, 26173848663000, 391153031777263, 6163682285356171, 102136840106457790, 1775499429402739247, 32307194057014483391 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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) not= 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

Galway, W. F., 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

R. P. Stanley, Permutations

FORMULA

Sum_{n=0..infinity} a(n)x^n = (1-x^2)^{-1/4} (1+x)^{1/2} sum_{k=0..infinity) 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.

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 *)

CROSSREFS

Cf. A000111, A115455.

Sequence in context: A005967 A104859 A108289 * A084161 A325294 A230960

Adjacent sequences:  A007776 A007777 A007778 * A007780 A007781 A007782

KEYWORD

nonn,nice,easy

AUTHOR

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

EXTENSIONS

Edited by Ralf Stephan, May 08 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 13 02:09 EDT 2020. Contains 336441 sequences. (Running on oeis4.)