cp's OEIS Frontend

A243664 Number of 3-packed words of degree n.

%I A243664 #42 Feb 29 2020 18:10:14
%S A243664 1,1,21,1849,426405,203374081,173959321557,242527666641289,
%T A243664 514557294036701349,1577689559404884503761,6714435826042791310638741,
%U A243664 38401291553086405072860452569,287412720357301174793668207559205,2753382861926383584939774967275568801
%N A243664 Number of 3-packed words of degree n.
%C A243664 See Novelli-Thibon (2014) for precise definition.
%H A243664 Andrew Howroyd, <a href="/A243664/b243664.txt">Table of n, a(n) for n = 0..100</a> (terms n = 0..30 from Peter Luschny)
%H A243664 J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 16.
%F A243664 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
%F A243664 a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n,3*k) * a(n-k). - _Ilya Gutkovskiy_, Jan 21 2020
%p A243664 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
%t A243664 g[t_] := (Exp[t] + 2 Exp[-t/2] Cos[Sqrt[3] t/2])/3;
%t A243664 a[n_] := (3n)! SeriesCoefficient[1/(2 - g[t^(1/3)]), {t, 0, n}];
%t A243664 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 13 2018, after _Peter Luschny_ *)
%o A243664 (Sage)
%o A243664 def CEN(m, len):
%o A243664     f, e, r, u = [1], [1], [1], 1
%o A243664     for i in (1..len-1):
%o A243664         f.append(rising_factorial(u, m))
%o A243664         for k in range(i-1, -1, -1):
%o A243664             e[k] = (e[k]*f[i])//f[i-k]
%o A243664         s = sum(e); e.append(s); r.append(s)
%o A243664         u += m
%o A243664     return r
%o A243664 A243664 = lambda len: CEN(3,len)
%o A243664 A243664(14) # _Peter Luschny_, Jul 06 2015
%o A243664 (Sage) # Alternative
%o A243664 def PackedWords3(n):
%o A243664     shapes = [[x**3 for x in p] for p in Partitions(n)]
%o A243664     return sum([factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes])
%o A243664 [PackedWords3(n) for n in (0..13)] # _Peter Luschny_, Aug 02 2015
%o A243664 (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
%Y A243664 Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.
%K A243664 nonn
%O A243664 0,3
%A A243664 _N. J. A. Sloane_, Jun 14 2014
%E A243664 a(0)=1 prepended, more terms from _Peter Luschny_, Jul 06 2015