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A292838
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Number of sets of nonempty words with a total of n letters over quaternary alphabet.
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3
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1, 4, 22, 132, 729, 4000, 21488, 113760, 594548, 3073392, 15732936, 79846448, 402104884, 2010879968, 9992425872, 49366096352, 242584319710, 1186177166680, 5773569726884, 27982357252632, 135079969593838, 649640609539360, 3113354757088720, 14871179093155424
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Product_{j>=1} (1+x^j)^(4^j).
a(n) ~ 4^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(4^(m-1)-1)) = 0.147762663788961720137665013823002812172... - Vaclav Kotesovec, Sep 28 2017
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MAPLE
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h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1)*binomial(4^i, j), j=0..n/i)))
end:
a:= n-> h(n$2):
seq(a(n), n=0..30);
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MATHEMATICA
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h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[h[n - i j, i - 1] Binomial[4^i, j], {j, 0, n/i}]]];
a[n_] := h[n, n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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