OFFSET
0,4
COMMENTS
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.4(a).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau, Tree-like Tableaux, Electronic Journal of Combinatorics, 20(4), 2013, #P34.
Venkatesan Guruswami, Enumerative aspects of certain subclasses of perfect graphs, Discrete Math. 205 (1999), 97-117. See Th. 6.3.
Sam Spiro, Counting Threshold Graphs with Eulerian Numbers, arXiv:1909.06518 [math.CO], 2019.
Sam Spiro, Subset Parking Functions, arXiv:1909.10109 [math.CO], 2019.
FORMULA
a(n) = c(n) - n*c(n-1) where c() = A000670.
a(n) ~ n!/2 * (1-log(2))/(log(2))^(n+1). - Vaclav Kotesovec, Dec 08 2012
Binomial transform is A005840. - Michael Somos, Aug 01 2016
a(n) = Sum_{k=0..n-1} binomial(n, k) * a(k), n>1. - Michael Somos, Aug 01 2016
a(n) = A005840(n) / 2, n>1. - Michael Somos, Aug 01 2016
E.g.f. A(x) satisfies (1 - x) * A'(x) = A(x) * (x - 2 + 2*A(x)). - Michael Somos, Aug 01 2016
EXAMPLE
G.f. = 1 + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1437*x^6 + 14512*x^7 + ...
MAPLE
A053525 := proc(n) option remember;
`if`(n < 2, 1 - n, add(binomial(n, k) * A053525(k), k = 0..n-1)) end:
seq(A053525(n), n = 0..20); # Peter Luschny, Oct 24 2021
MATHEMATICA
With[{nn=20}, CoefficientList[Series[(1-x)/(2-Exp[x]), {x, 0, nn}], x] Range[0, nn]!] Harvey P. Dale, May 17 2012
PROG
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 - x) / (2 - exp(x + x*O(x^n))), n))}; /* Michael Somos, Aug 01 2016 */
(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
(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
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane, Jan 15 2000
STATUS
approved