A014304 Expansion of e.g.f. 1/sqrt(exp(x)*(2-exp(x))).

1, 0, 1, 3, 16, 105, 841, 7938, 86311, 1062435, 14605306, 221790723, 3687263581, 66609892440, 1299237505021, 27213601303983, 609223983928576, 14516520372130245, 366820998284761861, 9798039716677045218

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

			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
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..420
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
• S. Linusson and J. Shareshian, Complexes of t-colorable graphs, SIAM J. Discrete Math., 16(3), 371-389. (19 pages).
• Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function, arXiv:2110.08576 [math.CO], 2021.

Programs

Magma

m:=30; R:=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

Maple

a := n -> -add((-1)^(n-k)*Stirling2(n+1, k)*doublefactorial(2*k-3), k = 0..n+1):
seq(a(n), n = 0..19); # Peter Luschny, Oct 19 2021

Mathematica

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 *)

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

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

Formula

a(n) = Sum_{k=0..n} ((2*k)!/k!)^2*Stirling2(n,2*k)/4^k. - Vladimir Kruchinin, Jan 19 2012

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