OFFSET
0,3
LINKS
Fung Lam, Table of n, a(n) for n = 0..394
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019.
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (14) with r = 2.
FORMULA
E.g.f.: (1 - x - sqrt(1-2*x-x^2))/x.
E.g.f. A(x) satisfies x*A(x)^2 + 2*(x-1)*A(x) + 2*x = 0, A(0)=0 and A(x) = x/(1-x-(x/2)*A(x)). - Michael Somos, Sep 06 2003
a(n) = n!*Sum_{k=0..floor((n-1)/2)} binomial(n-1, 2k)*binomial(2k, k)/(2^k*(k+1)). - Emanuele Munarini, Feb 06 2013
a(n) ~ sqrt(2-sqrt(2))*n^(n-1)/(exp(n)*(sqrt(2)-1)^(n+1)). - Vaclav Kotesovec, Sep 24 2013
Recurrence: (n+1)*a(n) = n*(n-1)*(n-2)*a(n-2) + n*(2*n-1)*a(n-1), n >= 3, a(1)=1, a(2)=2. - Fung Lam, Feb 24 2014
a(n) = n!*hypergeom([(1 - n)/2, 1 - n/2], [2], 2) for n >= 1. Peter Luschny, Apr 20 2020
MAPLE
# This is a crude Maple program based on Eq. (14), p. 4, in Takacs (1993). It calculates a(n) for n >= 2. Here, r = 2 is the maximum out-degree of each node.
ff := proc(r, n) simplify(subs(x = 0, diff(sum(x^k/k!, k = 0 .. r)^n, x$(n - 1)))); end;
seq(ff(2, i), i = 2 .. 40); # Petros Hadjicostas, Jun 09 2019
MATHEMATICA
Range[0, 20]! CoefficientList[Series[(1 - x - ((x - 1)^2 - 2 x^2)^(1/2))/x, {x, 0, 20}], x] (* Geoffrey Critzer, Nov 22 2011 *)
f[r_, n_][x_] := Sum[x^k/k!, {k, 0, r}]^n;
a[n_] := If[n == 1, 1, Derivative[n - 1][f[2, n]][0]];
a /@ Range[0, 19] (* Jean-François Alcover, Apr 20 2020, after Petros Hadjicostas *)
a[n_] := n! Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2, 2]; a[0] = 0;
Array[a, 20, 0] (* Peter Luschny, Apr 20 2020 *)
PROG
(PARI) a(n)=if(n<1, 0, n!*polcoeff(2*x/(1-x+sqrt(1-2*x-x^2+O(x^n))), n))
(PARI) a(n)=if(n<1, 0, n!*polcoeff(serreverse(2*x/(2+2*x+x^2)+x*O(x^n)), n))
(Maxima) makelist(n!*sum(binomial(n-1, 2*k)*binomial(2*k, k)/(2^k*(k+1)), k, 0, floor((n-1)/2)), n, 0, 20); /* Emanuele Munarini, Feb 06 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better description and formula from Christian G. Bower, Nov 29 2001
"unordered" added to the name by David Callan, Apr 22 2012
STATUS
approved