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%I #190 Mar 29 2024 11:43:54
%S 1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,36,36,46,46,60,60,74,74,94,
%T 94,114,114,140,140,166,166,202,202,238,238,284,284,330,330,390,390,
%U 450,450,524,524,598,598,692,692,786,786,900,900,1014,1014,1154,1154,1294,1294
%N Binary partition function: number of partitions of n into powers of 2.
%C First differences of A000123; also A000123 with terms repeated. See the relevant proof that follows the first formula below.
%C Among these partitions there is exactly one partition with all distinct terms, as every number can be expressed as the sum of the distinct powers of 2.
%C Euler transform of A036987 with offset 1.
%C a(n) is the number of "non-squashing" partitions of n, that is, partitions n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k. - _N. J. A. Sloane_, Nov 30 2003
%C Normally the OEIS does not include sequences like this where every term is repeated, but an exception was made for this one because of its importance. The unrepeated sequence A000123 is the main entry.
%C Number of different partial sums from 1 + [1, *2] + [1, *2] + ..., where [1, *2] means we can either add 1 or multiply by 2. E.g., a(6) = 6 because we have 6 = 1 + 1 + 1 + 1 + 1 + 1 = (1+1) * 2 + 1 + 1 = 1 * 2 * 2 + 1 + 1 = (1+1+1) * 2 = 1 * 2 + 1 + 1 + 1 + 1 = (1*2+1) * 2 where the connection is defined via expanding each bracket; e.g., this is 6 = 1 + 1 + 1 + 1 + 1 + 1 = 2 + 2 + 1 + 1 = 4 + 1 + 1 = 2 + 2 + 2 = 2 + 1 + 1 + 1 + 1 = 4 + 2. - _Jon Perry_, Jan 01 2004
%C Number of partitions p of n such that the number of compositions generated by p is odd. For proof see the Alekseyev and Adams-Watters link. - _Vladeta Jovovic_, Aug 06 2007
%C Differs from A008645 first at a(64). - _R. J. Mathar_, May 28 2008
%C Appears to be row sums of A155077. - _Mats Granvik_, Jan 19 2009
%C Number of partitions (p_1, p_2, ..., p_k) of n, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i >= p_{i+1} + ... + p_k. - John MCKAY (mckay(AT)encs.concordia.ca), Mar 06 2009 (these are the "non-squashing" partitions as nonincreasing lists).
%C Equals rightmost diagonal of triangle of A168261. Starting with offset 1 = eigensequence of triangle A115361 and row sums of triangle A168261. - _Gary W. Adamson_, Nov 21 2009
%C Equals convolution square root of A171238: (1, 2, 5, 8, 16, 24, 40, 56, 88, ...). - _Gary W. Adamson_, Dec 05 2009
%C Let B = the n-th convolution power of the sequence and C = the aerated variant of B. It appears that B/C = the binomial sequence beginning (1, n, ...). Example: Third convolution power of the sequence is (1, 3, 9, 19, 42, 78, 146, ...), with C = (1, 0, 3, 0, 9, 0, 19, ...). Then B/C = (1, 3, 6, 10, 15, 21, ...). - _Gary W. Adamson_, Aug 15 2016
%C From _Gary W. Adamson_, Sep 08 2016: (Start)
%C The limit of the matrix power M^k as n-->inf results in a single column vector equal to the sequence, where M is the following production matrix:
%C 1, 0, 0, 0, 0, ...
%C 1, 0, 0, 0, 0, ...
%C 1, 1, 0, 0, 0, ...
%C 1, 1, 0, 0, 0, ...
%C 1, 1, 1, 0, 0, ...
%C 1, 1, 1, 0, 0, ...
%C 1, 1, 1, 1, 0, ...
%C 1, 1, 1, 1, 0, ...
%C 1, 1, 1, 1, 1, ...
%C ... (End)
%C a(n) is the number of "non-borrowing" partitions of n, meaning binary subtraction of a smaller part from a larger part will never require place-value borrowing. - _David V. Feldman_, Jan 29 2020
%H T. D. Noe, <a href="/A018819/b018819.txt">Table of n, a(n) for n = 0..1000</a>
%H Max Alekseyev and Franklin T. Adams-Watters, <a href="/A018819/a018819.txt">Two proofs of an observation of Vladeta Jovovic</a>
%H Giedrius Alkauskas, <a href="http://uosis.mif.vu.lt/~alkauskas/MP3/rodseth.pdf"> Generalization of the Rodseth-Gupta theorem on binary partitions</a>, Lithuanian Math. J., 43 (2) (2003), 103-110.
%H Giedrius Alkauskas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Alkauskas/alkauskas2.html">Congruence Properties of the Function that Counts Compositions into Powers of 2 </a>, J. Int. Seq. 13 (2010), 10.5.3.
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 38.1, p.729.
%H Scott M. Bailey and Donald M. Larson, <a href="https://arxiv.org/abs/2107.01316">The A(1)-module structure of the homology of Brown-Gitler spectra</a>, arXiv:2107.01316 [math.AT], 2021.
%H Valentin P. Bakoev, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00096-7">Algorithmic approach to counting of certain types m-ary partitions</a>, Discrete Mathematics, 275 (2004) pp. 17-41.
%H Philippe Biane, <a href="https://arxiv.org/abs/1810.00548">Laver tables and combinatorics</a>, arXiv:1810.00548 [math.CO], 2018. Mentions this sequence.
%H Peter J. Cameron, Firdous Ee Jannat, Rajat Kanti Nath, and Reza Sharafdini, <a href="https://arxiv.org/abs/2403.09423">A survey on conjugacy class graphs of groups</a>, arXiv:2403.09423 [math.GR], 2024.
%H Karl Dilcher and Larry Ericksen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Dilcher/dilcher44.html">Polynomial Analogues of Restricted b-ary Partition Functions</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.2.
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 48, 581.
%H Maciej Gawron, Piotr Miska and Maciej Ulas, <a href="https://arxiv.org/abs/1703.01955">Arithmetic properties of coefficients of power series expansion of Prod_{n>=0} (1-x^(2^n))^t</a>, arXiv:1703.01955 [math.NT], 2017.
%H Michael D. Hirschhorn and James A. Sellers, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper110.pdf">A different view of m-ary partitions</a>, Australasian J. Combin., vol.30 (2004), 193-196.
%H Jonathan Jordan and Richard Southwell, <a href="http://dx.doi.org/10.4236/am.2010.15045">Further Properties of Reproducing Graphs</a>, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From _N. J. A. Sloane_, Feb 03 2013
%H Yasuyuki Kachi and Pavlos Tzermias, <a href="http://mi.mathnet.ru/adm508">On the m-ary partition numbers</a>, Algebra and Discrete Mathematics, Volume 19 (2015). Number 1, pp. 67-76.
%H Matjaž Konvalinka and Igor Pak, <a href="http://www.math.ucla.edu/~pak/papers/CayleyComp7.pdf">Cayley compositions, partitions, polytopes, and geometric bijections</a>, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91; see also <a href="https://doi.org/10.1016/j.jcta.2013.11.008">DOI link</a>. - From _N. J. A. Sloane_, Dec 22 2012
%H Apisit Pakapongpun and Thomas Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ward/ward17.html">Functorial Orbit counting</a>, J. Int. Seq., 12 (2009) 09.2.4, example 25.
%H Øystein J. Rodseth and James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Sellers/sellers75.html">On a Restricted m-Non-Squashing Partition Function</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
%H David Ruelle, <a href="http://www.ams.org/notices/200208/fea-ruelle.pdf">Dynamical zeta functions and transfer operators</a>, Notices Amer. Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.
%H N. J. A. Sloane and James A. Sellers, <a href="http://arxiv.org/abs/math/0312418">On non-squashing partitions</a>, arXiv:math/0312418 [math.CO], 2003.
%H N. J. A. Sloane and James A. Sellers, <a href="http://dx.doi.org/10.1016/j.disc.2004.11.014">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%F a(2m+1) = a(2m), a(2m) = a(2m-1) + a(m). Proof: If n is odd there is a part of size 1; removing it gives a partition of n - 1. If n is even either there is a part of size 1, whose removal gives a partition of n - 1, or else all parts have even sizes and dividing each part by 2 gives a partition of n/2.
%F G.f.: 1 / Product_{j>=0} (1-x^(2^j)).
%F a(n) = (1/n)*Sum_{k = 1..n} A038712(k)*a(n-k), n > 1, a(0) = 1. - _Vladeta Jovovic_, Aug 22 2002
%F a(2*n) = a(2*n + 1) = A000123(n). - _Michael Somos_, Aug 25 2003
%F a(n) = 1 if n = 0, Sum_{j = 0..floor(n/2)} a(j) if n > 0. - _David W. Wilson_, Aug 16 2007
%F G.f. A(x) satisfies A(x^2) = (1-x) * A(x). - _Michael Somos_, Aug 25 2003
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*v^2 + v^3. - _Michael Somos_, Apr 10 2005
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6 * u1^3 - 3*u3*u2*u1^2 + 3*u3*u2^2*u1 - u3*u2^3. - _Michael Somos_, Oct 15 2006
%F G.f.: 1/( Sum_{n >= 0} x^evil(n) - x^odious(n) ), where evil(n) = A001969(n) and odious(n) = A000069(n). - _Paul D. Hanna_, Jan 23 2012
%F Let A(x) by the g.f. and B(x) = A(x^k), then 0 = B*((1-A)^k - (-A)^k) + (-A)^k, see fxtbook link. - _Joerg Arndt_, Dec 17 2012
%F G.f.: Product_{n>=0} (1+x^(2^n))^(n+1), see the fxtbook link. - _Joerg Arndt_, Feb 28 2014
%F G.f.: 1 + Sum_{i>=0} x^(2^i) / Product_{j=0..i} (1 - x^(2^j)). - _Ilya Gutkovskiy_, May 07 2017
%e G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 6*x^7 + 10*x^8 + ...
%e a(4) = 4: the partitions are 4, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.
%e a(7) = 6: the partitions are 4 + 2 + 1, 4 + 1 + 1 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1.
%e From _Joerg Arndt_, Dec 17 2012: (Start)
%e The a(10) = 14 binary partitions of 10 are (in lexicographic order)
%e [ 1] [ 1 1 1 1 1 1 1 1 1 1 ]
%e [ 2] [ 2 1 1 1 1 1 1 1 1 ]
%e [ 3] [ 2 2 1 1 1 1 1 1 ]
%e [ 4] [ 2 2 2 1 1 1 1 ]
%e [ 5] [ 2 2 2 2 1 1 ]
%e [ 6] [ 2 2 2 2 2 ]
%e [ 7] [ 4 1 1 1 1 1 1 ]
%e [ 8] [ 4 2 1 1 1 1 ]
%e [ 9] [ 4 2 2 1 1 ]
%e [10] [ 4 2 2 2 ]
%e [11] [ 4 4 1 1 ]
%e [12] [ 4 4 2 ]
%e [13] [ 8 1 1 ]
%e [14] [ 8 2 ]
%e The a(11) = 14 binary partitions of 11 are obtained by appending 1 to each partition in the list.
%e The a(10) = 14 non-squashing partitions of 10 are (in lexicographic order)
%e [ 1] [ 6 3 1 1 ]
%e [ 2] [ 6 3 2 ]
%e [ 3] [ 6 4 1 ]
%e [ 4] [ 6 5 ]
%e [ 5] [ 7 2 1 1 ]
%e [ 6] [ 7 2 2 ]
%e [ 7] [ 7 3 1 ]
%e [ 8] [ 7 4 ]
%e [ 9] [ 8 2 1 ]
%e [10] [ 8 3 ]
%e [11] [ 9 1 1 ]
%e [12] [ 9 2 ]
%e [13] [ 10 1 ]
%e [14] [ 11 ]
%e The a(11) = 14 non-squashing partitions of 11 are obtained by adding 1 to the first part in each partition in the list.
%e (End)
%e From _David V. Feldman_, Jan 29 2020: (Start)
%e The a(10) = 14 non-borrowing partitions of 10 are (in lexicographic order)
%e [ 1] [1 1 1 1 1 1 1 1 1 1]
%e [ 2] [2 2 2 2 2]
%e [ 3] [3 1 1 1 1 1 1 1]
%e [ 4] [3 3 1 1 1 1]
%e [ 5] [3 3 2 2]
%e [ 6] [3 3 3 1]
%e [ 7] [5 1 1 1 1 1]
%e [ 8] [5 5]
%e [ 9] [6 2 2]
%e [10] [6 4]
%e [11] [7 1 1 1]
%e [12] [7 3]
%e [13] [9 1]
%e [14] [10]
%e The a(11) = 14 non-borrowing partitions of 11 are obtained either by adding 1 to the first even part in each partition (if any) or else appending a 1 after the last part.
%e (End)
%e For example, the five partitions of 4, written in nonincreasing order, are [1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]. The last four satisfy the condition, and a(4) = 4. The Maple program below verifies this for small values of n.
%p with(combinat); N:=8; a:=array(1..N); c:=array(1..N);
%p for n from 1 to N do p:=partition(n); np:=nops(p); t:=0;
%p for s to np do r:=p[s]; r:=sort(r,`>`); nr:=nops(r); j:=1;
%p # while j<nr and r[j]>sum(r[k],k=j+1..nr) do j:=j+1;od; # gives A040039
%p while j<nr and r[j]>= sum(r[k],k=j+1..nr) do j:=j+1;od; # gives A018819
%p if j=nr then t:=t+1;fi od; a[n]:=t; od; # John McKay
%t max = 59; a[0] = a[1] = 1; a[n_?OddQ] := a[n] = a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + a[n/2]; Table[a[n], {n, 0, max}]
%t (* or *) CoefficientList[Series[1/Product[(1-x^(2^j)), {j, 0, Log[2, max] // Ceiling}], {x, 0, max}], x] (* _Jean-François Alcover_, May 17 2011, updated Feb 17 2014 *)
%t a[ n_] := If[n<1, Boole[n==0], a[n] = a[n-1] + If[EvenQ@n, a[Quotient[n,2]], 0]]; (* _Michael Somos_, May 04 2022 *)
%o (PARI) { n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*2)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } /* _Jon Perry_ */
%o (PARI) {a(n) = my(A, m); if( n<1, n==0, m=1; A = 1 + O(x); while(m<=n, m*=2; A = subst(A, x, x^2) / (1 - x)); polcoeff(A, n))}; /* _Michael Somos_, Aug 25 2003 */
%o (PARI) {a(n) = if( n<1, n==0, if( n%2, a(n-1), a(n/2)+a(n-1)))}; /* _Michael Somos_, Aug 25 2003 */
%o (Haskell)
%o a018819 n = a018819_list !! n
%o a018819_list = 1 : f (tail a008619_list) where
%o f (x:xs) = (sum $ take x a018819_list) : f xs
%o -- _Reinhard Zumkeller_, Jan 28 2012
%o (Haskell)
%o import Data.List (intersperse)
%o a018819 = (a018819_list !!)
%o a018819_list = 1 : 1 : (<*>) (zipWith (+)) (intersperse 0) (tail a018819_list)
%o -- _Johan Wiltink_, Nov 08 2018
%o (Python)
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def A018819(n): return 1 if n == 0 else A018819(n-1) + (0 if n % 2 else A018819(n//2)) # _Chai Wah Wu_, Jan 18 2022
%Y A000123 is the main entry for the binary partition function and gives many more properties and references.
%Y Cf. A115625 (labeled binary partitions), A115626 (labeled non-squashing partitions).
%Y Convolution inverse of A106400.
%Y Cf. A023893, A062051, A105420, A131995, A040039, A018819, A088567, A089054, A115361, A168261, A171238, A179051, A008619.
%K nonn,nice,easy
%O 0,3
%A _David W. Wilson_, _N. J. A. Sloane_ and _J. H. Conway_
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