OFFSET
1,3
COMMENTS
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
REFERENCES
D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..300
D. Chataur and M. Livernet, Adem-Cartan operads, arXiv:math/0209363 [math.AT], 2002-2003; Communications in Algebra 33 (2005), 4337-4360.
A. Giorgilli and G. Molteni, Representation of a 2-power as sum of k 2-powers: a recursive formula, J. Number Theory 133 (2013), no. 4, 1251-1261.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
J. Ježek and T. Kepka, Free entropic groupoids, Commentationes Mathematicae Universitatis Carolinae,Tome 022 (1981), p. 223-233.
D. E. Knuth, Letter to R. E. Tarjan & N. J. A. Sloane, Jul. 1975
Daniel Krenn and Stephan Wagner, Compositions into Powers of b : Asymptotic Enumeration and Parameters, arXiv:1410.4331 [math.NT], 2014.
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
G. Molteni, Cancellation in a short exponential sum, J. Number Theory 130 (2010), no. 9, 2011-2027.
G. Molteni, Representation of a 2-power as sum of k 2-powers: the asymptotic behavior, Int. J. Number Theory 8 (2012), no. 8, 1923-1963.
FORMULA
a(n) = coefficient of z^(2^n) in (z+z^2+z^4+...+z^(2^n))^n. - Don Knuth.
From Giuseppe Molteni, Dec 14 2012: (Start)
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
a(n) = A323840(n,n). - Alois P. Heinz, Mar 31 2021
EXAMPLE
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.
MAPLE
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):
seq(a(n), n=1..23); # Alois P. Heinz, Nov 07 2017
MATHEMATICA
f[n_] := Coefficient[Expand[Sum[z^(2^j), {j, n}]^n], z, 2^n]; Array[f, 20] (* Robert G. Wilson v, Apr 08 2012 *)
PROG
(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)}
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Hugo van der Sanden
Minor edits from Vaclav Kotesovec, Jul 26 2014
STATUS
approved