0,4

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

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.

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

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

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 *)

(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

Main diagonal of A378561 (shifted).

Sequence in context: A074518 A200317 A202295 * A202293 A202294 A029767

Adjacent sequences: A088786 A088787 A088788 * A088790 A088791 A088792

nonn,nice

Paul D. Hanna, Oct 15 2003

More terms from Alois P. Heinz, Aug 14 2008

Minor edits by Vaclav Kotesovec, Mar 31 2014

approved