A243665 Number of 4-packed words of degree n.

1, 1, 71, 35641, 65782211, 323213457781, 3482943541940351, 72319852680213967921, 2637329566270689344838491, 157544683317273333844553610061, 14601235867276343036803577794300631, 2010110081536549910297353731858747088201, 396647963186245408341324212422008625649510771

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

1/(2-(cos(t^(1/4))+cosh(t^(1/4)))/2): series(%,t,14): seq((4*n)!*coeff(%,t,n),n=0..12); # Peter Luschny, Jul 07 2015

Mathematica

g[t_] := (Cos[t] + Cosh[t])/2;
a[n_] := (4n)! SeriesCoefficient[1/(2 - g[t^(1/4)]), {t, 0, n}];
Table[a[n], {n, 0, 15}] (* 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(4*n, 4*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020

Sage

# uses[CEN from A243664]
A243665 = lambda len: CEN(4,len)
A243665(13) # Peter Luschny, Jul 06 2015

Sage

# Alternatively:
def PackedWords4(n):
    shapes = ([x*4 for x in p] for p in Partitions(n))
    return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
[PackedWords4(n) for n in (0..12)] # Peter Luschny, Aug 02 2015

Formula

a(n) = (4*n)! * [t^n] 1/(2-g(t^(1/4))) with g(t) = (cos(t) + cosh(t))/2. - Peter Luschny, Jul 07 2015

a(0) = 1; a(n) = Sum_{k=1..n} binomial(4*n,4*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020

Extensions

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