A088789 E.g.f.: REVERT(2*x/(1+exp(x))) = Sum_{n>=0} a(n)*x^n/n!.

0, 1, 1, 3, 14, 90, 738, 7364, 86608, 1173240, 17990600, 308055528, 5826331440, 120629547584, 2713659864832, 65909241461760, 1718947213795328, 47912968352783232, 1421417290991105664, 44717945211445216640, 1487040748881346835200, 52117255681017313721088

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

Sequence in context: A074518 A200317 A202295 * A202293 A202294 A029767

Adjacent sequences: A088786 A088787 A088788 * A088790 A088791 A088792

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