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A053525 Expansion of e.g.f.: (1-x)/(2-exp(x)).

%I A053525 #63 Dec 10 2024 05:39:44
%S A053525 1,0,1,4,23,166,1437,14512,167491,2174746,31374953,497909380,
%T A053525 8619976719,161667969646,3265326093109,70663046421208,
%U A053525 1631123626335707,40004637435452866,1038860856732399105,28476428717448349996,821656049857815980455
%N A053525 Expansion of e.g.f.: (1-x)/(2-exp(x)).
%C A053525 The number of connected labeled threshold graphs on n vertices. - _Sam Spiro_, Sep 22 2019
%C A053525 Also the number of 2-interval parking functions of size n. - _Sam Spiro_, Sep 24 2019
%D A053525 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.4(a).
%H A053525 T. D. Noe, <a href="/A053525/b053525.txt">Table of n, a(n) for n = 0..100</a>
%H A053525 Jean-Christophe Aval, Adrien Boussicault, and Philippe Nadeau, <a href="https://doi.org/10.37236/3440">Tree-like Tableaux</a>, Electronic Journal of Combinatorics, 20(4), 2013, #P34.
%H A053525 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 A053525 Sam Spiro, <a href="https://arxiv.org/abs/1909.06518">Counting Threshold Graphs with Eulerian Numbers</a>, arXiv:1909.06518 [math.CO], 2019.
%H A053525 Sam Spiro, <a href="https://arxiv.org/abs/1909.10109">Subset Parking Functions</a>, arXiv:1909.10109 [math.CO], 2019.
%F A053525 a(n) = c(n) - n*c(n-1) where c() = A000670.
%F A053525 a(n) ~ n!/2 * (1-log(2))/(log(2))^(n+1). - _Vaclav Kotesovec_, Dec 08 2012
%F A053525 Binomial transform is A005840. - _Michael Somos_, Aug 01 2016
%F A053525 a(n) = Sum_{k=0..n-1} binomial(n, k) * a(k), n>1. - _Michael Somos_, Aug 01 2016
%F A053525 a(n) = A005840(n) / 2, n>1. - _Michael Somos_, Aug 01 2016
%F A053525 E.g.f. A(x) satisfies (1 - x) * A'(x) = A(x) * (x - 2 + 2*A(x)). - _Michael Somos_, Aug 01 2016
%F A053525 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 A053525 G.f. = 1 + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1437*x^6 + 14512*x^7 + ...
%p A053525 A053525 := proc(n) option remember;
%p A053525 `if`(n < 2, 1 - n, add(binomial(n, k) * A053525(k), k = 0..n-1)) end:
%p A053525 seq(A053525(n), n = 0..20); # _Peter Luschny_, Oct 24 2021
%t A053525 With[{nn=20},CoefficientList[Series[(1-x)/(2-Exp[x]),{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, May 17 2012 *)
%o A053525 (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 A053525 (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 A053525 (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 A053525 Cf. A000670, A005840, A052882.
%K A053525 nonn,nice,easy
%O A053525 0,4
%A A053525 _N. J. A. Sloane_, Jan 15 2000