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A243665
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Number of 4-packed words of degree n.
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9
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1, 1, 71, 35641, 65782211, 323213457781, 3482943541940351, 72319852680213967921, 2637329566270689344838491, 157544683317273333844553610061, 14601235867276343036803577794300631, 2010110081536549910297353731858747088201, 396647963186245408341324212422008625649510771
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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(n) = (4*n)! * [t^n] 1/(2-g(t^(1/4))) with g(t) = (cos(t) + cosh(t))/2. - Peter Luschny, Jul 07 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(4*n,4*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020
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MAPLE
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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
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MATHEMATICA
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g[t_] := (Cos[t] + Cosh[t])/2;
a[n_] := (4n)! SeriesCoefficient[1/(2 - g[t^(1/4)]), {t, 0, n}];
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PROG
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(Sage) # Alternatively:
def PackedWords4(n):
shapes = ([x*4 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(4*n, 4*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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