Search: id:A018819
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%I A018819
%S A018819 1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,36,36,46,46,60,60,74,74,94,94,
%T A018819 114,114,140,140,166,166,202,202,238,238,284,284,330,330,390,390,450,450,
%U A018819 524,524,598,598,692,692,786,786,900,900,1014,1014,1154,1154,1294,1294
%N A018819 Binary partition function: number of partitions of n into powers of 2.
%C A018819 First differences of A000123; also A000123 with terms repeated.
%C A018819 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 A018819 Euler transform of A036987 with offset 1.
%C A018819 a(n) = 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 (njas(AT)research.att.com), 
               Nov 30, 2003
%C A018819 Normally the OEIS does not include sequences like this where every terms 
               is repeated, but an exception was made for this one because of its 
               importance. The unrepeated sequence A000123 is the main entry.
%C A018819 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 
               (perry(AT)globalnet.co.uk), Jan 01 2004
%C A018819 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 (vladeta(AT)eunet.rs), Aug 06 2007
%C A018819 Differs from A008645 first at a(64). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               May 28 2008
%C A018819 Appears to be row sums of A155077. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Jan 19 2009]
%C A018819 Comment from John MCKAY (mckay(AT)encs.concordia.ca), Mar 06 2009 (Start): 
               Apparently a(n) = 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.
%C A018819 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2009: 
               (Start)
%C A018819 Equals rightmost diagonal of triangle of A168261. Starting with offset 
               1
%C A018819 = eigensequence of triangle A115361 and row sums of triangle A168261. 
               (End)
%D A018819 O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition 
               Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
%D A018819 D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. 
               Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.
%H A018819 T. D. Noe, <a href="b018819.txt">Table of n, a(n) for n=0..1000</a>
%H A018819 Max Alekseyev and Franklin T. Adams-Watters, <a href="a018819.txt">Two 
               proofs of an observation of Vladeta Jovovic</a>
%H A018819 M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, 
               <a href="http://www.lacim.uqam.ca/~plouffe/OEIS/archive_in_pdf/mike-m-ary.pdf">
               Australasian J. Combin.</a>, 30 (2004), 193-196.
%H A018819 M. D. Hirschhorn and J. A. Sellers, <a href="http://www.math.psu.edu/
               sellersj/mike-m-ary.pdf">A different view of m-ary partitions</a>
%H A018819 N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/
               0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 
               259-274.
%H A018819 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               48, 581
%F A018819 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 A018819 G.f.: 1 / Product_{j=0..inf} (1-x^(2^j)).
%F A018819 a(n)=(1/n)*Sum_{k=1..n} A038712(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Aug 22 2002
%F A018819 a(n) = 1 if n = 0, Sum(j = 0..[n/2], a(j)) if n > 0. - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 16 2007
%F A018819 G.f. A(x) satisfies A(x^2)=(1-x)A(x). - Michael Somos, Aug 25 2003
%F A018819 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2w -2uv^2 
               +v^3. - Michael Somos Apr 10 2005
%F A018819 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
%e A018819 a(4) = 4: the partitions are 4, 2+2, 2+1+1, 1+1+1+1; 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
%p A018819 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 A018819 (Maple code from John McKay) with(combinat); N:=8; a:=array(1..N); c:=array(1..N);
%p A018819 for n from 1 to N do p:=partition(n); np:=nops(p); t:=0;
%p A018819 for s to np do r:=p[s]; r:=sort(r,`>`); nr:=nops(r); j:=1;
%p A018819 #while j<nr and r[j]>sum(r[k],k=j+1..nr) do j:=j+1;od; # gives A040039
%p A018819 while j<nr and r[j]>= sum(r[k],k=j+1..nr) do j:=j+1;od; # gives A018819
%p A018819 if j=nr then t:=t+1;fi od; a[n]:=t; od;
%o A018819 (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 A018819 (PARI) {a(n)=local(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 Apr 10 
               2005 */
%o A018819 (PARI) {a(n)=if(n<1, n==0, if(n%2, a(n-1), a(n/2)+a(n-1)))}
%Y A018819 A000123(n)=a(2n)=a(2n+1). A000123 is the main entry for the binary partition 
               function and gives many more properties and references.
%Y A018819 Cf. A115625 (labeled binary partitions), A115626 (labeled non-squashing 
               partitions).
%Y A018819 Convolution inverse of A106400.
%Y A018819 Cf. A023893, A062051, A105420, A131995, A040039.
%Y A018819 Cf. A018819, A088567, A089054.
%Y A018819 Sequence in context: A008643 A008644 A008645 this_sequence A127370 A106247 
               A094909
%Y A018819 Cf. A115361, A168261 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 
               21 2009]
%Y A018819 Adjacent sequences: A018816 A018817 A018818 this_sequence A018820 A018821 
               A018822
%K A018819 nonn,nice,easy,new
%O A018819 0,3
%A A018819 David W. Wilson (davidwwilson(AT)comcast.net), N. J. A. Sloane (njas(AT)research.att.com) 
               and J. H. Conway (conway(AT)math.princeton.edu)

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