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A243666 Number of 5-packed words of degree n. 6

%I

%S 1,1,253,762763,11872636325,633287284180541,90604069581412784683,

%T 29529277377602939454694793,19507327717978242212109900308085,

%U 23927488379043876045061553841299192011,50897056444296458534155179226333868898628813,177758773838827813873239281786548960244155096117573

%N Number of 5-packed words of degree n.

%C See Novelli-Thibon (2014) for precise definition.

%H 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 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 a(0) = 1; a(n) = Sum_{k=1..n} binomial(5*n,5*k) * a(n-k). - _Ilya Gutkovskiy_, Jan 21 2020

%p 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 b = (5 - Sqrt[5])/4; c = (5 + Sqrt[5])/4;

%t 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 a[n_] := (5n)! SeriesCoefficient[1/(2 - g[t]), { t, 0, 5 n}] // Simplify;

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* _Jean-Fran├žois Alcover_, Jul 14 2018, after _Peter Luschny_ *)

%o (Sage) # uses[CEN from A243664]

%o A243666 = lambda len: CEN(5,len)

%o A243666(12) # _Peter Luschny_, Jul 06 2015

%o (Sage) # Alternatively:

%o def PackedWords5(n):

%o shapes = ([x*5 for x in p] for p in Partitions(n))

%o return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)

%o [PackedWords5(n) for n in (0..11)] # _Peter Luschny_, Aug 02 2015

%o (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 Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 14 2014

%E a(0)=1 prepended, more terms from _Peter Luschny_, Jul 06 2015

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Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)