Search: id:A007779
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%I A007779
%S A007779 1,1,1,2,5,17,72,367,2179,14750,112023,942879,8708912,87563937,
%T A007779 951933849,11125383714,139092236301,1852257089937,26173848663000,
%U A007779 391153031777263,6163682285356171,102136840106457790
%N A007779 Coefficients of asymptotic expansion of Ramanujan false theta series.
%C A007779 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. - R. P. Stanley (rstan(AT)math.mit.edu), Jan 22 2006. 
               For example, a(3)=2 because there are two alternating fixed-point-free 
               involutions on 1,...,6, viz., 214365 and 645231.
%C A007779 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). - R. P. Stanley (rstan(AT)math.mit.edu), 
               Jan 22 2006
%D A007779 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 545.
%H A007779 Galway, W. F., <a href="http://www.math.uiuc.edu/Reports/galway/97-020.html">
               An Asymptotic Expansion of Ramanujan</a>, 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.
%H A007779 R. P. Stanley, <a href="http://arXiv.org/abs/math.CO/0603520">Alternating 
               permutations and symmetric functions</a>
%F A007779 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)) - R. P. Stanley (rstan(AT)math.mit.edu), Jan 22 2006
%F A007779 Berndt gives an explicit g.f. on page 547.
%Y A007779 Cf. A115455.
%Y A007779 Sequence in context: A005967 A104859 A108289 this_sequence A084161 A102038 
               A002135
%Y A007779 Adjacent sequences: A007776 A007777 A007778 this_sequence A007780 A007781 
               A007782
%K A007779 nonn,nice,easy
%O A007779 0,4
%A A007779 William F. Galway [ galway(AT)math.uiuc.edu ]
%E A007779 Edited by Ralf Stephan, May 08 2007

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