A087803 Number of unlabeled rooted trees with at most n nodes.

1, 2, 4, 8, 17, 37, 85, 200, 486, 1205, 3047, 7813, 20299, 53272, 141083, 376464, 1011311, 2732470, 7421146, 20247374, 55469206, 152524387, 420807242, 1164532226, 3231706871, 8991343381, 25075077710, 70082143979, 196268698287, 550695545884, 1547867058882

Offset: 1

Author

Hugo Pfoertner, Oct 12 2003

Keywords

• nonn

Comments

Number of equations (order conditions) that must be satisfied to achieve order n in the construction of a Runge-Kutta method for the numerical solution of an ordinary differential equation. - Hugo Pfoertner, Oct 12 2003

References

• Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, (1987) Wiley, Chichester
• See link for more references.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
• I. Th. Famelis, S. N. Papakostas and Ch. Tsitouras, Symbolic Derivation of Runge-Kutta Order Conditions.
• R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy).
• R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
• Florian Ingels, Revisiting Tree Isomorphism: An Algorithmic Bric-à-Brac, arXiv:2309.14441 [cs.DS], 2023-2024. See p. 17.
• Eric Weisstein's World of Mathematics, Rooted Tree.
• Index entries for sequences related to rooted trees.

Crossrefs

a(n) = Sum_(k=1..n) A000081(k).

Cf. A255170, A187770, A051491.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; b(n) +`if`(n<1, 0, a(n-1)) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 21 2012

Mathematica

b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[b[n - j]* DivisorSum[j, # *b[#]&], {j, 1, n-1}]/(n-1); a[1] = 1; a[n_] := a[n] = b[n] + a[n-1]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)
t[1] = a[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n}, {m, (n-1)/k}]; a[n_] := a[n] = a[n-1] + t[n]; Table[a[n], {n, 40}] (* Vladimir Reshetnikov, Aug 12 2016 *)
Needs["NumericalDifferentialEquationAnalysis`"]
Drop[Accumulate[Join[{0},ButcherTreeCount[20]]],1] (* Peter Luschny, Aug 18 2016 *)

PARI

a000081(k) = local(A = x); if( k<1, 0, for( j=1, k-1, A /= (1 - x^j + x * O(x^k))^polcoeff(A, j)); polcoeff(A, k));
a(n) = sum(k=1, n, a000081(k)) \\ Altug Alkan, Nov 10 2015

Sage

def A087803_list(len):
    a, t = [1], [0,1]
    for n in (1..len-1):
        S = [t[n-k+1]*sum(d*t[d] for d in divisors(k)) for k in (1..n)]
        t.append(sum(S)//n)
        a.append(a[-1]+t[-1])
    return a
A087803_list(20) # Peter Luschny, Aug 18 2016

Formula

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.664861031240097088000569... . - Vaclav Kotesovec, Sep 11 2014

In the asymptotics above the constant c = A187770 / (1 - 1 / A051491). - Vladimir Reshetnikov, Aug 12 2016

Extensions

Corrected and extended by Alois P. Heinz, Aug 21 2012

Renamed (old name is in comments) by Vladimir Reshetnikov, Aug 23 2016