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Expansion of e.g.f. 1/sqrt(exp(x)*(2-exp(x))).
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%I #40 Sep 08 2022 08:44:39

%S 1,0,1,3,16,105,841,7938,86311,1062435,14605306,221790723,3687263581,

%T 66609892440,1299237505021,27213601303983,609223983928576,

%U 14516520372130245,366820998284761861,9798039716677045218

%N Expansion of e.g.f. 1/sqrt(exp(x)*(2-exp(x))).

%C a(n) is the absolute value of the reduced Euler characteristic for the simplicial complex of bipartite (2-colorable) graphs on n+1 vertices. - Jakob Jonsson (jonsson(AT)mathematik.uni-marburg.de), Apr 03 2003

%C F(x) = -sqrt(2*exp(-x) - 1) + 1 is the exponential generating function for the sequence shifted one step (0,1,0,1,3,16,105,...). The first derivative of F coincides with the generating function in the name line. F aligns better with the given Euler characteristic as the coefficient of x^n corresponds to graphs on n vertices (rather than n+1 vertices). - Jakob Jonsson (jonsson(AT)mathematik.uni-marburg.de), Apr 03 2003

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.5.

%H G. C. Greubel, <a href="/A014304/b014304.txt">Table of n, a(n) for n = 0..420</a>

%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2011-2013.

%H S. Linusson and J. Shareshian, <a href="http://www.math.wustl.edu/~shareshi/papers.html">Complexes of t-colorable graphs</a>, SIAM J. Discrete Math., 16(3), 371-389. (19 pages).

%H Feng Qi and Mark Daniel Ward, <a href="https://arxiv.org/abs/2110.08576">Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function</a>, arXiv:2110.08576 [math.CO], 2021.

%F a(n) = Sum_{k=0..n} ((2*k)!/k!)^2*Stirling2(n,2*k)/4^k. - _Vladimir Kruchinin_, Jan 19 2012

%F a(n) ~ n^n / (sqrt(2) * log(2)^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Jan 11 2017

%F a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jun 11 2020

%F a(n) = -Sum_{k=0..n+1} (-1)^(n-k)*Stirling2(n+1,k)*(2*k-3)!! (see Qi/Ward). - _Peter Luschny_, Oct 19 2021

%e a(3) = 3 because the following graphs are bipartite on four vertices: The empty graph (1 graph); all graphs with one edge (6 graphs); all graphs with two edges (15 graphs); graphs with three edges not forming a triangle (16 graphs); and graphs with four edges forming a square (3 graphs). The reduced Euler characteristic is hence -1 + 6 - 15 + 16 - 3 = 3. - Jakob Jonsson (jonsson(AT)mathematik.uni-marburg.de), Apr 03 2003

%p a := n -> -add((-1)^(n-k)*Stirling2(n+1, k)*doublefactorial(2*k-3), k = 0..n+1):

%p seq(a(n), n = 0..19); # _Peter Luschny_, Oct 19 2021

%t With[{nn=30},CoefficientList[Series[1/Sqrt[Exp[x](2-Exp[x])],{x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Dec 12 2012 *)

%o (Maxima) a(n):=sum(((2*k)!/k!)^2*stirling2(n,2*k)/4^k,k,0,n); /* _Vladimir Kruchinin_, Jan 19 2012 */

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( 1/sqrt(exp(x)*(2-exp(x))) )) \\ _G. C. Greubel_, Jun 12 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Sqrt(Exp(x)*(2-Exp(x))) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jun 12 2019

%o (Sage) m = 30; T = taylor(1/sqrt(exp(x)*(2-exp(x))), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Jun 12 2019

%K nonn

%O 0,4

%A _N. J. A. Sloane_