OFFSET
1,3
COMMENTS
a(n) is also the number of permutation graphs with domination number one. See Definition 2.1, Lemma 2.3, and page 16 in the paper provided in the link by Theresa Baren, et al. - Daniel A. McGinnis, Oct 16 2018
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49.
K. Wayland, personal communication.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, and Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.
Todd Feil, Gary Kennedy and David Callan, Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]
FORMULA
a(n) ~ 2 * (n-1)! * (1 - 1/(2*n) + 1/(2*n^2) + 9/(2*n^3) + 59/(2*n^4) + 237/n^5 + 2280/n^6 + 25182/n^7 + 625385/(2*n^8) + 4311329/n^9 + 65375943/n^10). - Vaclav Kotesovec, Mar 17 2015
a(n) = Sum_{k=1..n} (n-k)!*A145878(k-1,0). See the link by Theresa Baren, et al. - Daniel A. McGinnis, Oct 15 2018
a(n) = A003149(n-1) - Sum_{k=0..n-1} (n-k-1)!*a(k). (This follows immediately from the preceding formula since A145878(k,0) = k! - a(k).) - Pontus von Brömssen, Jul 10 2021
a(n) + A052186(n) = n! - Pontus von Brömssen, Jul 10 2021
MAPLE
t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 1 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # James A. Sellers, Mar 13 2000
MATHEMATICA
m = 22; s = Sum[n!*x^n, {n, 0, m}]; Range[0, m-1]! - CoefficientList[ Series[ s/(1+x*s), {x, 0, m}], x][[1;; m]] // Rest (* Jean-François Alcover, Apr 28 2011, after Maple code *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Mar 13 2000
Edited by Emeric Deutsch, Oct 29 2008
STATUS
approved