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

Author

Paul D. Hanna, Oct 15 2003

Keywords

• nonn
• nice

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.

Crossrefs

Main diagonal of A378561 (shifted).

Programs

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 *)

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

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

Extensions

More terms from Alois P. Heinz, Aug 14 2008

Minor edits by Vaclav Kotesovec, Mar 31 2014