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A007178
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Number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats.
(Formerly M2951)
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13
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1, 1, 3, 13, 75, 525, 4347, 41245, 441675, 5259885, 68958747, 986533053, 15292855019, 255321427725, 4567457001915, 87156877087069, 1767115200924299, 37936303950503853, 859663073472084315, 20505904049009202685, 513593410566661282347, 13476082013068430626893
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OFFSET
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1,3
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COMMENTS
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Also the dimension of the arity n component of the operad of level algebras (see the reference by Chataur-Livernet by definition), and the cardinality of the subset of the free commutative medial magma with n generators that contains each generator exactly once. The linear operad of level algebras is the linearization of the set operad of commutative medial magmas; the statement about commutative medial magmas follows from the description in the paper of Ježek-Kepka. - Vladimir Dotsenko, Mar 12 2022
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REFERENCES
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D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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D. Chataur and M. Livernet, Adem-Cartan operads, arXiv:math/0209363 [math.AT], 2002-2003; Communications in Algebra 33 (2005), 4337-4360.
J. Ježek and T. Kepka, Free entropic groupoids, Commentationes Mathematicae Universitatis Carolinae,Tome 022 (1981), p. 223-233.
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FORMULA
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a(n) = coefficient of z^(2^n) in (z+z^2+z^4+...+z^(2^n))^n. - Don Knuth.
Limit_{n->oo} (a(n)/n!)^(1/n) = 1.192674341213466032221288982528755... (see References: "Representation of a 2-power as sum of k 2-powers: the asymptotic behavior").
a(n) == 4 + (-1)^n (mod 8) for n > 2 (see References: "Representation of a 2-power as sum of k 2-powers: a recursive formula"). (End)
More precise asymptotics: a(n) ~ c * d^n * n!, where d = 1.192674341213466032221288982528755176734489232027131552652821007498903522051783..., c = 0.24849369086953813603231092781945750388624874631949260927875431616785914609... - Vaclav Kotesovec, Sep 20 2019
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EXAMPLE
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For n=3, the 3 sums are 1/2 + 1/4 + 1/4, 1/4 + 1/2 + 1/4, and 1/4 + 1/4 + 1/2.
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MAPLE
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b:= proc(n, r, p) option remember; `if`(n<r, 0,
`if`(r=0, `if`(n=0, p!, 0), add(1/j!*
b(n-j, 2*(r-j), p+j), j=0..min(n, r))))
end:
a:= n-> b(n, 1, 0):
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MATHEMATICA
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f[n_] := Coefficient[Expand[Sum[z^(2^j), {j, n}]^n], z, 2^n]; Array[f, 20] (* Robert G. Wilson v, Apr 08 2012 *)
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PROG
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(PARI) f(n)={my(M); if(n>1, M=matrix(n, n); M[2, 1] = 1; for(k=3, n, for(l=1, k-2, M[k, l] = 0; smx = min(2*l, k-l-1); for(s=1, smx, M[k, l] += binomial(k+l-1, 2*l-s)*M[k-l, s])); M[k, k-1] = 1); M[n, 1], 1)}
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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