OFFSET
0,4
COMMENTS
a(n+1) is also number of ways to place n nonattacking composite pieces semi-rook + semi-bishop on an n X n board. Two semi-bishops (see A187235) do not attack each other if they are in the same northwest-southeast diagonal. Two semi-rooks do not attack each other if they are in the same column (see also semi-queens, A099152). - Vaclav Kotesovec, Dec 22 2011
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
V. Dotsenko, Pattern avoidance in labelled trees, arXiv preprint arXiv:1110.0844 [math.CO], 2011.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 716-719.
R. W. Whitty, Rook polynomials on two-dimensional surfaces and graceful labellings of graphs, Discrete Math., 308 (2008), 674-683.
FORMULA
E.g.f.: x/2 - LambertW(-x*exp(x/2)/2). - Vladeta Jovovic, Feb 12 2008
a(n) = (1/2^n)*Sum_{k=1..n} binomial(n,k)*k^(n-1)= A038049(n)/2^n, n>1. - Vladeta Jovovic, Feb 12 2008
Asymptotics: a(n)/(n-2)! ~ b * q^(n-1) * sqrt(n), where q = 1/(2*LambertW(1/exp(1))) = 1.795560738334311... is the root of the equation 2*q = exp(1+1/(2*q)) and b = 1/(2*LambertW(1/exp(1))) * sqrt((1+LambertW(1/exp(1)))/(2*Pi)) = 0.8099431005... - Vaclav Kotesovec, Dec 22 2011, updated Sep 25 2012
MAPLE
a:= n->coeff(series(x/2-LambertW(-1/2*x*exp(1/2*x)), x=0, n+1), x, n)*n!:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 14 2008
MATHEMATICA
Table[n!/2^n*Sum[2^j/j!*StirlingS2[n-1, n-j], {j, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Dec 25 2011 *)
With[{nmax = 50}, CoefficientList[Series[x/2 - LambertW[-x*Exp[x/2]/2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)
PROG
(PARI) a(n)=local(A); if(n<0, 0, A=x+O(x^n); n!*polcoeff(serreverse(2*x/(1 + exp(x))), n))
(PARI) x='x+O('x^50); concat([0], Vec(serlaplace(x/2 - lambertw(-x*exp(x/2)/2)))) \\ G. C. Greubel, Nov 14 2017
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Paul D. Hanna, Oct 15 2003
EXTENSIONS
More terms from Alois P. Heinz, Aug 14 2008
Minor edits by Vaclav Kotesovec, Mar 31 2014
STATUS
approved