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 A243666 Number of 5-packed words of degree n. 7
 1, 1, 253, 762763, 11872636325, 633287284180541, 90604069581412784683, 29529277377602939454694793, 19507327717978242212109900308085, 23927488379043876045061553841299192011, 50897056444296458534155179226333868898628813, 177758773838827813873239281786548960244155096117573 (list; graph; refs; listen; history; text; internal format)
 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(0) = 1; a(n) = Sum_{k=1..n} binomial(5*n,5*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020 MAPLE 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 MATHEMATICA b = (5 - Sqrt[5])/4; c = (5 + Sqrt[5])/4; 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; a[n_] := (5n)! SeriesCoefficient[1/(2 - g[t]), { t, 0, 5 n}] // Simplify; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *) PROG (Sage) # uses[CEN from A243664] A243666 = lambda len: CEN(5, len) A243666(12) # Peter Luschny, Jul 06 2015 (Sage) # Alternatively: def PackedWords5(n): shapes = ([x*5 for x in p] for p in Partitions(n)) return sum(factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes) [PackedWords5(n) for n in (0..11)] # 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(5*n, 5*k) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020 CROSSREFS Cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words of degree n for 0<=k<=5. Sequence in context: A253880 A145628 A237419 * A243687 A263380 A144855 Adjacent sequences: A243663 A243664 A243665 * A243667 A243668 A243669 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

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Last modified February 4 10:27 EST 2023. Contains 360053 sequences. (Running on oeis4.)