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A014304
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Expansion of e.g.f. 1/sqrt(exp(x)*(2-exp(x))).
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6
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1, 0, 1, 3, 16, 105, 841, 7938, 86311, 1062435, 14605306, 221790723, 3687263581, 66609892440, 1299237505021, 27213601303983, 609223983928576, 14516520372130245, 366820998284761861, 9798039716677045218
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OFFSET
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0,4
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COMMENTS
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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
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
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.5.
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LINKS
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FORMULA
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a(n) ~ n^n / (sqrt(2) * log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 11 2017
a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jun 11 2020
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
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EXAMPLE
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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
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MAPLE
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a := n -> -add((-1)^(n-k)*Stirling2(n+1, k)*doublefactorial(2*k-3), k = 0..n+1):
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MATHEMATICA
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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 *)
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PROG
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(Maxima) a(n):=sum(((2*k)!/k!)^2*stirling2(n, 2*k)/4^k, k, 0, n); /* Vladimir Kruchinin, Jan 19 2012 */
(PARI) my(x='x+O('x^30)); Vec(serlaplace( 1/sqrt(exp(x)*(2-exp(x))) )) \\ G. C. Greubel, Jun 12 2019
(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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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