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A088789
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E.g.f.: REVERT(2*x/(1+exp(x))) = sum(n>=0, a(n)*x^n/n!).
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4
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0, 1, 1, 3, 14, 90, 738, 7364, 86608, 1173240, 17990600, 308055528, 5826331440, 120629547584, 2713659864832, 65909241461760, 1718947213795328, 47912968352783232, 1421417290991105664, 44717945211445216640
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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
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REFERENCES
| V. Dotsenko, Pattern avoidance in labelled trees, Arxiv preprint arXiv:1110.0844, 2011
R. W. Whitty, Rook polynomials on two-dimensional surfaces..., Discrete Math., 308 (2008), 674-683.
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..100
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FORMULA
| E.g.f.: x/2-LambertW(-1/2*x*exp(1/2*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), Feb 12 2008
Asymptotics: a(n)/(n-2)! ~ b*q^(n-1)*sqrt(n), where q is the root 1.795560738334311... of the equation 2q=e^(1+1/(2q)) and b=0.8099431005... - Vaclav Kotesovec, Dec 22 2011
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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
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MATHEMATICA
| Table[n!/2^n*Sum[2^j/j!*StirlingS2[n-1, n-j], {j, 1, n}], {n, 1, 50}] (* From Vaclav Kotesovec, Dec 25 2011 *)
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PROG
| (PARI) a(n)=local(A); if(n<0, 0, A=x+O(x^n); n!*polcoeff(serreverse(2*x/(1+exp(x))), n))
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CROSSREFS
| Sequence in context: A074518 A200317 A202295 * A202293 A202294 A029767
Adjacent sequences: A088786 A088787 A088788 * A088790 A088791 A088792
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 15 2003
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008
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