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A001862

Number of forests of least height with n nodes.

1, 1, 2, 7, 26, 111, 562, 3151, 19252, 128449, 925226, 7125009, 58399156, 507222535, 4647051970, 44747776651, 451520086208, 4761032807937, 52332895618066, 598351410294193, 7102331902602676, 87365859333294151, 1111941946738682522, 14621347433458883187

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OFFSET

0,3

COMMENTS

From Gus Wiseman, Feb 14 2024: (Start)

Also the number of minimal loop-graphs covering n vertices. This is the minimal case of A322661. For example, the a(0) = 1 through a(3) = 7 loop-graphs are (loops represented as singletons):

{} {1} {12} {1-23}

{1-2} {2-13}

{3-12}

{12-13}

{12-23}

{13-23}

{1-2-3}

(End)

REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. See (3.3.7): number of ways to cover the complete graph K_n with star graphs.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

John Riordan, Letter to N. J. A. Sloane, Oct. 1970

John Riordan and N. J. A. Sloane, Correspondence, 1974

FORMULA

E.g.f.: exp(x*(exp(x)-x/2)).

Binomial transform of A053530. - Gus Wiseman, Feb 14 2024

MATHEMATICA

Range[0, 20]! CoefficientList[Series[Exp[x Exp[x] - x^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 13 2011 *)