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A243665 Number of 4-packed words of degree n.

%I A243665 #39 Mar 16 2020 05:38:40
%S A243665 1,1,71,35641,65782211,323213457781,3482943541940351,
%T A243665 72319852680213967921,2637329566270689344838491,
%U A243665 157544683317273333844553610061,14601235867276343036803577794300631,2010110081536549910297353731858747088201,396647963186245408341324212422008625649510771
%N A243665 Number of 4-packed words of degree n.
%C A243665 See Novelli-Thibon (2014) for precise definition.
%H A243665 Andrew Howroyd, <a href="/A243665/b243665.txt">Table of n, a(n) for n = 0..100</a> (terms n = 0..30 from Peter Luschny)
%H A243665 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 A243665 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
%F A243665 a(0) = 1; a(n) = Sum_{k=1..n} binomial(4*n,4*k) * a(n-k). - _Ilya Gutkovskiy_, Jan 21 2020
%p A243665 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
%t A243665 g[t_] := (Cos[t] + Cosh[t])/2;
%t A243665 a[n_] := (4n)! SeriesCoefficient[1/(2 - g[t^(1/4)]), {t, 0, n}];
%t A243665 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 14 2018, after _Peter Luschny_ *)
%o A243665 (Sage) # uses[CEN from A243664]
%o A243665 A243665 = lambda len: CEN(4,len)
%o A243665 A243665(13) # _Peter Luschny_, Jul 06 2015
%o A243665 (Sage) # Alternatively:
%o A243665 def PackedWords4(n):
%o A243665     shapes = ([x*4 for x in p] for p in Partitions(n))
%o A243665     return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
%o A243665 [PackedWords4(n) for n in (0..12)] # _Peter Luschny_, Aug 02 2015
%o A243665 (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
%Y A243665 Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.
%K A243665 nonn
%O A243665 0,3
%A A243665 _N. J. A. Sloane_, Jun 14 2014
%E A243665 a(0)=1 prepended, more terms from _Peter Luschny_, Jul 06 2015