A243664 Number of 3-packed words of degree n.

1, 1, 21, 1849, 426405, 203374081, 173959321557, 242527666641289, 514557294036701349, 1577689559404884503761, 6714435826042791310638741, 38401291553086405072860452569, 287412720357301174793668207559205, 2753382861926383584939774967275568801

Offset: 0

Author

N. J. A. Sloane, Jun 14 2014

Keywords

• nonn

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.

Crossrefs

Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.

Programs

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 *)

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

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

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

Extensions

a(0)=1 prepended, more terms from Peter Luschny, Jul 06 2015