A063881 Number of oriented trees rooted at an arc.

1, 4, 18, 80, 367, 1708, 8122, 39204, 191963, 950984, 4759626, 24030736, 122258314, 626162464, 3225926450, 16706775984, 86928097451, 454203897192, 2382255252398, 12537764465072, 66193294753768, 350472816969976, 1860542261745782, 9901018433270812

Offset: 2

Author

Vladeta Jovovic, Aug 27 2001

Keywords

• nonn

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 61, (3.3.7).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..100

Crossrefs

Cf. A000151, A000238.

Programs

Maple

B:= proc(n) option remember; if n<=1 then unapply(x,x) else unapply(convert(series(x*exp(2*sum(B(n-1)(x^k)/k, k=1..n-1)), x,n+1), polynom),x) fi end: a:= proc(n) local T; T:=B(n-1)(x); add(coeff(T,x,k)* coeff(T,x,n-k), k=1..n-1) end: seq(a(n), n=2..23); # Alois P. Heinz, Aug 23 2008

Mathematica

B[n_ /; n <= 1] = Identity; B[n_] := B[n] = Function[x, Evaluate[Normal[Series[x*Exp[2*Sum[B[n-1][x^k]/k, {k, 1, n-1}]], {x, 0, n+1}]]]]; a[n_] := Module[{T}, T = B[n-1][x]; Sum[Coefficient[T, x, k]*Coefficient[T, x, n-k], {k, 1, n-1}]]; Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

Formula

a(n) = A000151(n)- A000238(n). G.f.: A(x) = B(x)^2, where B(x) is g.f. for A000151.