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A243666
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Number of 5-packed words of degree n.
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7
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1, 1, 253, 762763, 11872636325, 633287284180541, 90604069581412784683, 29529277377602939454694793, 19507327717978242212109900308085, 23927488379043876045061553841299192011, 50897056444296458534155179226333868898628813, 177758773838827813873239281786548960244155096117573
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OFFSET
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0,3
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COMMENTS
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See Novelli-Thibon (2014) for precise definition.
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LINKS
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FORMULA
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a(0) = 1; a(n) = Sum_{k=1..n} binomial(5*n,5*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020
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MAPLE
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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
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MATHEMATICA
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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;
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PROG
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(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)
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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