A243666 Number of 5-packed words of degree n.

1, 1, 253, 762763, 11872636325, 633287284180541, 90604069581412784683, 29529277377602939454694793, 19507327717978242212109900308085, 23927488379043876045061553841299192011, 50897056444296458534155179226333868898628813, 177758773838827813873239281786548960244155096117573

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

a := (5+sqrt(5))/4: b := (5-sqrt(5))/4: g := t -> (exp(t)+2*exp(t-a*t)*cos(t*sqrt(b/2))+2*exp(t-b*t)*cos(t*sqrt(a/2)))/5: series(1/(2-g(t)),t,56): seq((5*n)!*(coeff(simplify(%),t,5*n)),n=0..11); # Peter Luschny, Jul 07 2015

Mathematica

b = (5 - Sqrt[5])/4; c = (5 + Sqrt[5])/4;
g[t_] := (Exp[t] + 2*Exp[t - c*t]*Cos[t*Sqrt[b/2]] + 2*Exp[t - b*t]* Cos[t*Sqrt[c/2]])/5;
a[n_] := (5n)! SeriesCoefficient[1/(2 - g[t]), { t, 0, 5 n}] // Simplify;
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Jul 14 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(5*n, 5*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020

Sage

# uses[CEN from A243664]
A243666 = lambda len: CEN(5,len)
A243666(12) # Peter Luschny, Jul 06 2015

Sage

# Alternatively:
def PackedWords5(n):
    shapes = ([x*5 for x in p] for p in Partitions(n))
    return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
[PackedWords5(n) for n in (0..11)] # Peter Luschny, Aug 02 2015

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(5*n,5*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020

Extensions

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