Sequence A023359 Initial Value

[Franklin T. Adams-Watters]

I think the first value for A023359 should be 1, not 0.  The number of partitions or compositions of zero is normally taken to be 1.  I don't see any reason for this sequence to take it to be zero.  The remainder of the entry can also be simplified if this change is made.

The modified entry would be as follows:

%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 permuta\
  tions 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.u\
  k), Jun 22 2003
%H A023359 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/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 instea\
  d 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 (abcloitre(AT)modulo\
  net.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). - Pa\
  ul 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 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)



-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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