Sequence A023359 Initial Value

[David Wilson]

Works for me, I don't know why I would have put a(0) = 0 anyway.

----- Original Message ----- 
From: "Franklin T. Adams-Watters" <franktaw at netscape.net>
To: <seqfan at ext.jussieu.fr>
Sent: Saturday, August 06, 2005 12:10 AM
Subject: Sequence A023359 Initial Value


>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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