OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n, 1*u_1+2*u_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...;u_1, u_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*... *1^t_1*t_1!*2^t_2*t_2!*... *1^u_1*u_1!*2^u_2*u_2!*...)) where fixA[...] = Product_{i, j>=1} ( (Sum_{d|lcm(i, j)} (d*u_d))^(gcd(i, j)*s_i*t_j)).
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[Function[p, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
A242095[n_, k_] := A242095[n, k] = With[{co = Coefficient, ex = Exponent}, Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]*co[s, x, i]*co[t, x, j]}, If[g == 0, 1, Sum[d*co[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, ex[t, x]}], {i, ex[s, x]}]/Product[i^co[u, x, i]*co[u, x, i]!, {i, ex[u, x]}]/Product[i^co[t, x, i]*co[t, x, i]!, {i, ex[t, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, ex[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]];
a[n_] := A242095[n, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 29 2023, after Alois P. Heinz in A242095 *)
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
@cached_function
def monprod(M):
p = Pol.one()
V = [m.variables() for m in M]
T = cartesian_product(V)
for t in T:
r = [Pol.varname_key(str(u))[1] for u in t]
j = [Pol(M[u[0]]).degree(u[1]) for u in enumerate(t)]
lcm_r = lcm(r)
p *= x[lcm_r]^(prod(r)/lcm_r*prod(j))
return p
@cached_function
def pol_isotop(n, k):
P = Z(n)
p = Pol.zero()
coeffs = P.coefficients()
mons = P.monomials()
C = cartesian_product(k*[mons])
Csorted = [tuple(sorted(u)) for u in C]
Cset = set(Csorted)
for c in Cset:
p += Csorted.count(c)*prod([coeffs[mons.index(u)] for u in c])*monprod(c)
return p
@cached_function
def rule_sub(r, m):
D = 0
for d in divisors(r):
try: D += d*m.degrees()[-d-1]
except: break
return D
def a(n, k=2):
P = Z(n)
coeffs = P.coefficients()
Q = pol_isotop(n, k)
inds = [Pol.varname_key(str(u))[1] for u in Q.variables()]
p = 0
for mon in enumerate(P.monomials()):
m = Pol(mon[1])
p += coeffs[mon[0]]*Q.subs({x[i]:rule_sub(i, m) for i in inds})
return p
# Philip Turecek, Jun 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Dec 17 2003
EXTENSIONS
a(9) from Alois P. Heinz, Aug 14 2014
STATUS
approved