OFFSET
0,3
LINKS
Philip Turecek, Table of n, a(n) for n = 0..10
FORMULA
a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = Product_{i, j, k>=1} ( (Sum_{d|lcm(i, j, k)} (d*s_d))^(s_i*s_j*s_k*lcm(i, j, k)/(i*j*k))).
a(n) is asymptotic to n^(n^3)/n!.
PROG
(Sage)
Pol.<x> = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
if n==0: return Pol.one()
return sum(x[k]*Z(n-k) for k in (1..n))/n
def a(n, k=3):
P = Z(n)
q = 0
coeffs = P.coefficients()
for mon in enumerate(P.monomials()):
m = Pol(mon[1])
p = 1
V = m.variables()
T = cartesian_product(k*[V])
Tsorted = [tuple(sorted(u)) for u in T]
Tset = set(Tsorted)
for t in Tset:
r = [Pol.varname_key(str(u))[1] for u in t]
j = [m.degree(u) for u in t]
D = 0
lcm_r = lcm(r)
for d in divisors(lcm_r):
try: D += d*m.degrees()[-d-1]
except: break
p *= D^(Tsorted.count(t)*prod(r)/lcm_r*prod(j))
q += coeffs[mon[0]]*p
return q
# Philip Turecek, Jun 12 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Jan 16 2004
STATUS
approved