OFFSET
3,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 3..650
FORMULA
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = 0.13932434077355395... . - Vaclav Kotesovec, Sep 07 2014
EXAMPLE
a(5) = 3: .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:
g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
(add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
end:
p:= proc(n, i, t) option remember; `if`(n<t, 0, `if`(n=t, 1,
`if`(min(i, t)<1, 0, add(binomial(g(i)+j-1, j)*
p(n-i*j, i-1, t-j), j=0..min(n/i, t)))))
end:
a:= n-> p(n, n, 3):
seq(a(n), n=3..40);
MATHEMATICA
b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}]/(n - 1)];
g[n_] := g[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_, t_] := p[n, i, t] = If[n < t, 0, If[n == t, 1, If[Min[i, t] < 1, 0, Sum[Binomial[g[i] + j - 1, j]*p[n - i*j, i - 1, t - j], {j, 0, Min[n/i, t]}]]]];
a[n_] := p[n, n, 3];
a /@ Range[3, 40] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 29 2012
STATUS
approved