%I #60 Jan 01 2024 08:31:40

%S 1,0,1,4,23,166,1437,14512,167491,2174746,31374953,497909380,

%T 8619976719,161667969646,3265326093109,70663046421208,

%U 1631123626335707,40004637435452866,1038860856732399105,28476428717448349996,821656049857815980455

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

%C The number of connected labeled threshold graphs on n vertices. - _Sam Spiro_, Sep 22 2019

%C Also the number of 2-interval parking functions of size n. - _Sam Spiro_, Sep 24 2019

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.4(a).

%H T. D. Noe, <a href="/A053525/b053525.txt">Table of n, a(n) for n = 0..100</a>

%H Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p34">Tree-like Tableaux</a>, Electronic Journal of Combinatorics, 20(4), 2013, #P34.

%H Venkatesan Guruswami, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00022-9">Enumerative aspects of certain subclasses of perfect graphs</a>, Discrete Math. 205 (1999), 97-117. See Th. 6.3.

%H Sam Spiro, <a href="https://arxiv.org/abs/1909.06518">Counting Threshold Graphs with Eulerian Numbers</a>, arXiv:1909.06518 [math.CO], 2019.

%H Sam Spiro, <a href="https://arxiv.org/abs/1909.10109">Subset Parking Functions</a>, arXiv:1909.10109 [math.CO], 2019.

%F a(n) = c(n) - n*c(n-1) where c() = A000670.

%F a(n) ~ n!/2 * (1-log(2))/(log(2))^(n+1). - _Vaclav Kotesovec_, Dec 08 2012

%F Binomial transform is A005840. - _Michael Somos_, Aug 01 2016

%F a(n) = Sum_{k=0..n-1} binomial(n, k) * a(k), n>1. - _Michael Somos_, Aug 01 2016

%F a(n) = A005840(n) / 2, n>1. - _Michael Somos_, Aug 01 2016

%F E.g.f. A(x) satisfies (1 - x) * A'(x) = A(x) * (x - 2 + 2*A(x)). - _Michael Somos_, Aug 01 2016

%F a(n) = Sum_{k=1..n-1} (n-k)*A008292(n-1,k-1)*2^(k-1), valid for n>=2. - _Sam Spiro_, Sep 22 2019

%e G.f. = 1 + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1437*x^6 + 14512*x^7 + ...

%p A053525 := proc(n) option remember;

%p `if`(n < 2, 1 - n, add(binomial(n, k) * A053525(k), k = 0..n-1)) end:

%p seq(A053525(n), n = 0..20); # _Peter Luschny_, Oct 24 2021

%t With[{nn=20},CoefficientList[Series[(1-x)/(2-Exp[x]),{x,0,nn}], x] Range[0,nn]!] _Harvey P. Dale_, May 17 2012

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 - x) / (2 - exp(x + x*O(x^n))), n))}; /* _Michael Somos_, Aug 01 2016 */

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)/(2-Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Mar 15 2019

%o (Sage) m = 25; T = taylor((1-x)/(2-exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Mar 15 2019

%Y Cf. A000670, A005840, A052882.

%K nonn,nice,easy

%O 0,4

%A _N. J. A. Sloane_, Jan 15 2000