|
|
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) != 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
|
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.
|
|
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.
|
|
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
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
William F. Galway (galway(AT)math.uiuc.edu)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|