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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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6
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0, 1, 1, 3, 14, 90, 738, 7364, 86608, 1173240, 17990600, 308055528, 5826331440, 120629547584, 2713659864832, 65909241461760, 1718947213795328, 47912968352783232, 1421417290991105664, 44717945211445216640, 1487040748881346835200, 52117255681017313721088
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OFFSET
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0,4
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COMMENTS
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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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LINKS
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FORMULA
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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
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MAPLE
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a:= n->coeff(series(x/2-LambertW(-1/2*x*exp(1/2*x)), x=0, n+1), x, n)*n!:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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