OFFSET
0,3
COMMENTS
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..650
FORMULA
a(n) = A095133(2*n,n).
a(n) = A105821(2*n+1,n+1). - Alois P. Heinz, Jul 10 2013
a(n) = A136605(2*n+1,n). - Alois P. Heinz, Apr 11 2014
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.955765285..., c = 3.36695186... . - Vaclav Kotesovec, Sep 10 2014
EXAMPLE
a(0) = 1: ( ), the empty forest with 0 trees and 0 edges.
a(1) = 1: ( o-o ), 1 tree and 1 edge. o
a(2) = 2: ( o-o-o ), ( o-o o-o ). |
a(3) = 4: ( o-o-o-o ), ( o-o-o o-o ), ( o-o o-o o-o ), ( o-o-o ).
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:
t:= proc(n) option remember; local k; `if` (n=0, 1, b(n)-
(add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
`if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j)*
binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
end:
a:= n-> g(2*n, 2*n, n):
seq(a(n), n=0..40);
MATHEMATICA
nn = 30; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
a[1] = 1; sol =
SolveAlways[
0 == Series[
t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
b[x_] := Sum[a[n] x^n /. sol, {n, 0, nn}]; ft =
Drop[Flatten[
CoefficientList[Series[b[x] - (b[x]^2 - b[x^2])/2, {x, 0, nn}],
x]], 1]; Drop[
CoefficientList[
Series[Product[1/(1 - y ^(i - 1))^ft[[i]], {i, 2, nn}], {y, 0, nn}],
y], -1] (* Geoffrey Critzer, Nov 10 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 27 2012
STATUS
approved