A007178 Number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats.

1, 1, 3, 13, 75, 525, 4347, 41245, 441675, 5259885, 68958747, 986533053, 15292855019, 255321427725, 4567457001915, 87156877087069, 1767115200924299, 37936303950503853, 859663073472084315, 20505904049009202685, 513593410566661282347, 13476082013068430626893

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe, Don Knuth

Keywords

• nonn
• easy
• nice

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

Examples

			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.
		

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.

Crossrefs

Cf. A002572, A294746, A323840.

Programs

Maple

b:= proc(n, r, p) option remember; `if`(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 *)

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)}

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

Extensions

More terms from Hugo van der Sanden

Minor edits from Vaclav Kotesovec, Jul 26 2014