%I A023359
%S A023359 1,1,2,3,6,10,18,31,56,98,174,306,542,956,1690,2983,5272,9310,16448,
%T A023359 29050,51318,90644,160118,282826,499590,882468,1558798,2753448,4863696,
%U A023359 8591212,15175514,26805983,47350056,83639030,147739848,260967362
%N A023359 Number of ordered partitions of n into powers of 2.
%C A023359 a(n) is the number of partitions of 2n into n parts, with each partition 
               realized into non-symmetric permutations ignoring 1's. For example 
               a(6): the partitions of 12 into 6 are: 111117 (1), 111126 (1), 111135 
               (1), 111144 (1), 111225 (2), 111234 (3), 111333 (1), 112233 (3), 
               112224 (2), 122223 (2), 222222 (1), where the number in brackets 
               is the number of non-symmetric permutations ignoring 1's (e.g. 111234, 
               ignore 1's -> 234 and we can also have 243 and 324, 112233->2233 
               or 2323 or 2332). The sum of the bracketed numbers is a(6)=18. - 
               Jon Perry (perry(AT)globalnet.co.uk), Jun 22 2003
%C A023359 Equals right border of triangle A144219. Row sums of A144219 = (1, 2, 
               3, 6, 10, 18,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 14 2008]
%H A023359 T. D. Noe, <a href="b023359.txt">Table of n, a(n) for n=0..200</a>
%H A023359 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A023359 a(n) = (n=0) + sum(a(n-2^k), k=0...) - Len Smiley (smiley(AT)math.uaa.alaska.edu), 
               May 07 2001
%F A023359 A(x) = A(x^2)/(1 - x*A(x^2)). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Dec 16 2002
%F A023359 INVERT transform of characteristic function of powers of 2, i.e. A036987 
               interpreted with an offset 1 instead of 0. - Antti Karttunen, Dec 
               12, 2003
%F A023359 a(n) seems to be asymptotic to A*B^n where A=0.332198... B=1.766398... 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 17 2002
%F A023359 Satisfies A(x)=1+A(x)*sum(k>=0, x^(2^k)). a(m) == 1 (mod 2) when m=2^n-1, 
               otherwise a(m) == 0 (mod 2). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Aug 27 2003
%e A023359 A(x) = A(x^2) + x*A(x^2)^2 + x^2*A(x^2)^3 + x^3*A(x^2)^4 +... = 1 +x 
               +2x^2 +3x^3 +6x^4 + 10x^5 + 18x^6 +31x^7 +....
%o A023359 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=1/
               (1/subst(A,x,x^2)-x)); polcoeff(A,n))
%Y A023359 The column sums of the table A073265. Cf. also A073267, A073202, A073288.
%Y A023359 A144219 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 14 2008]
%Y A023359 Sequence in context: A102702 A060945 A077930 this_sequence A082482 A066000 
               A011957
%Y A023359 Adjacent sequences: A023356 A023357 A023358 this_sequence A023360 A023361 
               A023362
%K A023359 nonn,easy,nice
%O A023359 0,3
%A A023359 David W. Wilson (davidwwilson(AT)comcast.net)
%E A023359 Edited by Franklin T. Adams-Watters, Aug 05 2005