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A243664 Number of 3-packed words of degree n. 8
1, 1, 21, 1849, 426405, 203374081, 173959321557, 242527666641289, 514557294036701349, 1577689559404884503761, 6714435826042791310638741, 38401291553086405072860452569, 287412720357301174793668207559205, 2753382861926383584939774967275568801 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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

Sequence in context: A296597 A203327 A297652 * A232949 A305145 A296686

Adjacent sequences:  A243661 A243662 A243663 * A243665 A243666 A243667

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

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Last modified November 27 06:40 EST 2020. Contains 338678 sequences. (Running on oeis4.)