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A292805
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Number of sets of nonempty words with a total of n letters over n-ary alphabet.
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8
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1, 1, 5, 55, 729, 12376, 250735, 5904746, 158210353, 4747112731, 157545928646, 5726207734545, 226093266070501, 9632339536696943, 440262935648935344, 21482974431740480311, 1114363790702406540897, 61219233429920494716931, 3550130647865299090804375
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] Product_{j=1..n} (1+x^j)^(n^j).
a(n) ~ n^(n - 3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019
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EXAMPLE
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a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
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MAPLE
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h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
end:
a:= n-> h(n$3):
seq(a(n), n=0..20);
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MATHEMATICA
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h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
a[n_] := h[n, 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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