cp's OEIS Frontend

A243666 Number of 5-packed words of degree n.

%I A243666 #40 Mar 16 2020 05:37:37
%S A243666 1,1,253,762763,11872636325,633287284180541,90604069581412784683,
%T A243666 29529277377602939454694793,19507327717978242212109900308085,
%U A243666 23927488379043876045061553841299192011,50897056444296458534155179226333868898628813,177758773838827813873239281786548960244155096117573
%N A243666 Number of 5-packed words of degree n.
%C A243666 See Novelli-Thibon (2014) for precise definition.
%H A243666 Andrew Howroyd, <a href="/A243666/b243666.txt">Table of n, a(n) for n = 0..100</a> (terms n = 0..30 from Peter Luschny)
%H A243666 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 A243666 a(0) = 1; a(n) = Sum_{k=1..n} binomial(5*n,5*k) * a(n-k). - _Ilya Gutkovskiy_, Jan 21 2020
%p A243666 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
%t A243666 b = (5 - Sqrt[5])/4; c = (5 + Sqrt[5])/4;
%t A243666 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;
%t A243666 a[n_] := (5n)! SeriesCoefficient[1/(2 - g[t]), { t, 0, 5 n}] // Simplify;
%t A243666 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* _Jean-François Alcover_, Jul 14 2018, after _Peter Luschny_ *)
%o A243666 (Sage) # uses[CEN from A243664]
%o A243666 A243666 = lambda len: CEN(5,len)
%o A243666 A243666(12) # _Peter Luschny_, Jul 06 2015
%o A243666 (Sage) # Alternatively:
%o A243666 def PackedWords5(n):
%o A243666     shapes = ([x*5 for x in p] for p in Partitions(n))
%o A243666     return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
%o A243666 [PackedWords5(n) for n in (0..11)] # _Peter Luschny_, Aug 02 2015
%o A243666 (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
%Y A243666 Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.
%K A243666 nonn
%O A243666 0,3
%A A243666 _N. J. A. Sloane_, Jun 14 2014
%E A243666 a(0)=1 prepended, more terms from _Peter Luschny_, Jul 06 2015