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A007779 Coefficients of asymptotic expansion of Ramanujan false theta series.

%I A007779 #27 Aug 03 2024 07:10:20
%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 391153031777263,6163682285356171,102136840106457790,1775499429402739247,32307194057014483391
%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) != 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.
%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). - _Richard Stanley_, Jan 22 2006
%D A007779 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 545.
%H A007779 Alois P. Heinz, <a href="/A007779/b007779.txt">Table of n, a(n) for n = 0..200</a>
%H A007779 W. F. Galway, <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="https://arxiv.org/abs/math/0603520">Alternating permutations and symmetric functions</a>, arXiv:math/0603520 [math.CO], 2006.
%H A007779 R. P. Stanley, <a href="http://www.ams.org/amsmtgs/colloq-10.pdf">Permutations</a>
%F A007779 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
%F A007779 Berndt gives an explicit g.f. on page 547.
%t A007779 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 *)
%Y A007779 Cf. A000111, A115455.
%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