A000238 Number of oriented trees with n nodes.

1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, 492180, 2266502, 10598452, 50235931, 240872654, 1166732814, 5702001435, 28088787314, 139354922608, 695808554300, 3494390057212, 17641695461662, 89495023510876, 456009893224285, 2332997330210440

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, r(x).
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
• 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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 350 terms from N. J. A. Sloane)
• H. R. Afshar, E. A. Bergshoeff, W. Merbis, Interacting spin-2 fields in three dimensions, arXiv preprint arXiv:1410.6164 [hep-th], 2014-2015, JHEP 2015 (2015) # 040.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy), TOC
• R. J. Mathar, Oriented trees A000238
• R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
• Index entries for sequences related to trees

Crossrefs

Cf. A000060, A000151, A051437 (linear oriented), A334827 (oriented star-like).

Diagonal of A335362.

Programs

Maple

A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(2* add(A(n-1)(x^k)/k, k=1..n-1)), x=0,n), polynom), x) fi end: a:= n-> coeff(series(A(n+1)(x) *(1-A(n+1)(x)), x=0, n+1), x,n): seq(a(n), n=1..26); # Alois P. Heinz, Aug 20 2008

Mathematica

A[n_][y_] := A[n][y] = If[n == 0, 0, Normal[Series[x*Exp[2*Sum[A[n-1][x^k]/k, {k, 1, n-1}]], {x, 0, n}] /. x -> y]]; a[n_] := SeriesCoefficient[A[n+1][x]*(1-A[n+1][x]), {x, 0, n}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

PARI

seq(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); Vec(Ser(A)-x*Ser(A)^2)} \\ Andrew Howroyd, May 13 2018

Formula

G.f. = x+x^2+3*x^3+8*x^4+27*x^5+... = R(x)-R(x)^2, where R(x) = g.f. for A000151.

a(n) ~ c * d^n / n^(5/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.22571615379282714232305... . - Vaclav Kotesovec, Dec 08 2014

Extensions

2 errors corrected by Paul Zimmermann, Mar 01 1996

More terms from N. J. A. Sloane, Mar 10 2007