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) = (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
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 *)
PROG
(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
(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
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