A000123 and company

Wouter Meeussen eu000949 at pophost.eunet.be
Sat Aug 14 17:09:40 CEST 1999

Summary (for Neil):
A042949 is identical to A018819,
A033485 is Rest[A000123] /2

A000123 is (in short) :

ID Number: A000123 (Formerly M1011 and N0378)
Sequence:  1,2,4,6,10,14,20,26,36,46,60,74,94,114,140,166,202,238,284,..
Formula:   G.f.: (1-x)^{-1} {prod from { n =0 } to inf ( 1 - x^{2^n} )^{-1} .
Name:      Binary partitions (partitions of 2n into powers of 2):
See also:  Cf. A042949.
the pointer to A042949 leads to:
ID Number: A042949
Sequence:  1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,36,36,46,46,60,60,..       
Name:      First differences of A000123; also A000123 doubled up.

        IS SAME AS:

ID Number: A018819
Sequence:  1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,36,36,46,46,60,60,..
Name:      Partitions of n into powers of 2.
Formula:   G.f.: 1 / PROD (1-x^(2^j)).
Keywords:  nonn
Offset:    0
Author(s): David W. Wilson (wilson at ctron.com)

Now consider :
ID Number: A033485
Sequence:  1,2,3,5,7,10,13,18,23,30,37,47,57,70,83,101,119,142,165,195,
Name:      a(n)=a(n-1)+a(n/2) if n even, = a(n-1)+a((n-1)/2) if n odd.
See also:  Cf. A040039.
Keywords:  nonn,nice
Offset:    1
Author(s): Philippe Deleham, BP 29, Coconi, 97670 Ouangani, Mayotte.

ID Number: A040039
Sequence:  1,1,2,2,3,3,5,5,7,7,10,10,13,13,18,18,23,23,30,30,37,37,47,
Name:      First differences of A033485; also A033485 doubled up.
Keywords:  nonn
Offset:    0
Author(s): njas, jhc

neither have a GF or a link to A000123.
The first (=A033485) is (Rest[A000123])/2

[[an evidence :
[[ multiplying the GF(z) of a sequence with (1-z)
[[ produces the GF of the first differences of that sequence.
[[ dividing the GF(z) of a sequence by (1-z)
[[ produces the GF of the running sum of that sequence.

a propos,
the GF of A042949 and A018819 is
1/PROD[1-z^(2^j), {j,0,oo}]

with PROD[1-z^(2^j), {j,0,oo}] being the GF of
the parity of the number of 1's in the binary decomposition of n.

Is there a "simple" relation between the sequences generated by
GF and 1/GF ??

In the best of worlds, we ought to be able to look up the GF itself; but the
multitude of languages (Maple, Mma, C++) spoils it.


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