OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Partition of a set
FORMULA
a(n) = A124424(2n,n).
Conjecture: Limit_{n->oo} (a(n)/n!)^(1/n) = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773... - Vaclav Kotesovec, Oct 21 2023
EXAMPLE
a(2) = 5: 13|24, 14|2|3, 1|2|34, 1|23|4, 12|3|4.
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, expand(x*
add(g(n-j)*binomial(n-1, j-1), j=1..n)))
end:
S:= (n, k)-> coeff(g(n), x, k):
b:= proc(g, u) option remember;
add(S(g, k)*S(u, k)*k!, k=0..min(g, u))
end:
T:= proc(n, k) option remember; local g, u; g:= floor(n/2); u:= ceil(n/2);
add(add(add(binomial(g, i)*S(i, h)*binomial(u, j)*
S(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
end:
a:= n-> T(2*n, n):
seq(a(n), n=0..18);
MATHEMATICA
b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}];
T[n_, k_] := Module[{g, u}, g = Floor[n/2]; u = Ceiling[n/2]; Sum[Sum[Sum[ Binomial[g, i]*StirlingS2[i, h]*Binomial[u, j]*StirlingS2[j, k - h]*b[g - i, u - j], {j, k - h, u}], {i, h, g}], {h, 0, k}]];
a[n_] := T[2n, n];
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 01 2023
STATUS
approved