OFFSET
0,3
COMMENTS
See Novelli-Thibon (2014) for precise definition.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100 (terms n = 0..30 from Peter Luschny)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 16.
FORMULA
a(n) = (3*n)! * [t^n] 1/(2-g(t^(1/3))) with g(t) = (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3. - Peter Luschny, Jul 07 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n,3*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020
MAPLE
g := t -> (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3: series(1/(2-g(t^(1/3))), t, 14): seq(((3*n)!*coeff(%, t, n)), n=0..13); # Peter Luschny, Jul 07 2015
MATHEMATICA
g[t_] := (Exp[t] + 2 Exp[-t/2] Cos[Sqrt[3] t/2])/3;
a[n_] := (3n)! SeriesCoefficient[1/(2 - g[t^(1/3)]), {t, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 13 2018, after Peter Luschny *)
PROG
(Sage)
def CEN(m, len):
f, e, r, u = [1], [1], [1], 1
for i in (1..len-1):
f.append(rising_factorial(u, m))
for k in range(i-1, -1, -1):
e[k] = (e[k]*f[i])//f[i-k]
s = sum(e); e.append(s); r.append(s)
u += m
return r
A243664 = lambda len: CEN(3, len)
A243664(14) # Peter Luschny, Jul 06 2015
(Sage) # Alternative
def PackedWords3(n):
shapes = [[x**3 for x in p] for p in Partitions(n)]
return sum([factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes])
[PackedWords3(n) for n in (0..13)] # Peter Luschny, Aug 02 2015
(PARI) seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(3*n, 3*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 14 2014
EXTENSIONS
a(0)=1 prepended, more terms from Peter Luschny, Jul 06 2015
STATUS
approved