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A014304 Expansion of e.g.f. 1/sqrt(exp(x)*(2-exp(x))). 6
1, 0, 1, 3, 16, 105, 841, 7938, 86311, 1062435, 14605306, 221790723, 3687263581, 66609892440, 1299237505021, 27213601303983, 609223983928576, 14516520372130245, 366820998284761861, 9798039716677045218 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.5.
LINKS
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).
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
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
Sequence in context: A085614 A215931 A271777 * A370092 A063548 A157452
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified February 28 06:33 EST 2024. Contains 370387 sequences. (Running on oeis4.)