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A243664
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Number of 3-packed words of degree n.
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10
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1, 1, 21, 1849, 426405, 203374081, 173959321557, 242527666641289, 514557294036701349, 1577689559404884503761, 6714435826042791310638741, 38401291553086405072860452569, 287412720357301174793668207559205, 2753382861926383584939774967275568801
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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) = (3*n)! * [t^n] 1/(2-g(t^(1/3))) with g(t) = (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3. - Peter Luschny, Jul 07 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n,3*k) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020
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MAPLE
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g := t -> (exp(t)+2*exp(-t/2)*cos(sqrt(3)*t/2))/3: series(1/(2-g(t^(1/3))), t, 14): seq(((3*n)!*coeff(%, t, n)), n=0..13); # Peter Luschny, Jul 07 2015
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MATHEMATICA
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g[t_] := (Exp[t] + 2 Exp[-t/2] Cos[Sqrt[3] t/2])/3;
a[n_] := (3n)! SeriesCoefficient[1/(2 - g[t^(1/3)]), {t, 0, n}];
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PROG
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(Sage)
def CEN(m, len):
f, e, r, u = [1], [1], [1], 1
for i in (1..len-1):
f.append(rising_factorial(u, m))
for k in range(i-1, -1, -1):
e[k] = (e[k]*f[i])//f[i-k]
s = sum(e); e.append(s); r.append(s)
u += m
return r
(Sage) # Alternative
def PackedWords3(n):
shapes = [[x**3 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(3*n, 3*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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