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A087803
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Number of unlabeled rooted trees with at most n nodes.
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30
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, (1987) Wiley, Chichester
See link for more references.
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.664861031240097088000569... . - Vaclav Kotesovec, Sep 11 2014
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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)
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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