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A011800
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Number of labeled forests of n nodes each component of which is a path.
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5
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1, 1, 2, 7, 34, 206, 1486, 12412, 117692, 1248004, 14625856, 187638716, 2614602112, 39310384192, 634148436104, 10923398137576, 200069534481616, 3882002527006352, 79535575126745632, 1715658099715217584
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OFFSET
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0,3
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COMMENTS
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Denote the bivariate exponential g.f. by g(x,y)=exp(y*f(x)) where f(x)=(2x-x^2)/(2-2x). Then this sequence is the row sums of the array defined by the g.f. The differential dg/dy = f(x)*exp(y*f(x)) is the exponential generating function for an array with row sums in A201720. - R. J. Mathar, Jun 27 2022
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REFERENCES
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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (3.3.6).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(d).
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LINKS
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FORMULA
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E.g.f.: exp[ x + x^2/(2*(1 - x)) ].
Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-1)^2*a(n-2) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012
a(n) = n!*Sum_{k=1..n} (Sum_{i=0..n-k} binomial(k,n-k-i)*binomial(k+i-1,k-1)*2^(-n+k+i)*(-1)^(n-k-i))/k!, n > 0, a(0) = 1. - Vladimir Kruchinin, Nov 25 2012
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MATHEMATICA
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Function[ esl, esl*Array[ Factorial, Length[ esl ], 0 ] ][ CoefficientList[ Series[ Exp[ x+x^2/(2-2x) ], {x, 0, 20} ], x ] ] (* Olivier Gérard *)
With[{nn=20}, CoefficientList[Series[Exp[x+x^2/(2*(1-x))], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, May 13 2019 *)
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PROG
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(Maxima)
a(n):=n!*sum(sum(binomial(k, n-k-i)*binomial(k+i-1, k-1)*2^(-n+k+i)*(-1)^(n-k-i), i, 0, n-k)/(k!), k, 1, n); /* Vladimir Kruchinin, Nov 25 2012 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Herbert S. Wilf
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STATUS
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approved
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