login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000929 Dimension of n-th degree part of Steenrod algebra. 54
1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 20, 22, 23, 26, 28, 29, 32, 35, 37, 41, 45, 47, 51, 55, 58, 63, 68, 72, 77, 82, 86, 92, 98, 103, 111, 118, 123, 131, 139, 145, 154, 164, 171, 180, 190, 198, 208, 219, 229, 241, 253, 264, 278, 291 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of partitions p(1)+p(2)+...+p(m) = n (into positive parts) such that 2*p(k) <= p(k-1).

Number of partitions of n into parts of the form 2^j-1, j=1,2,... (called s-partitions). Example: a(7)=4 because we have [7], [3,3,1], [3,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 06 2006

One direction of a bijection between both sorts of partitions, as an algorithm: take a partition P (p(1)+p(2)+...+p(m) such that 2*p(k) <= p(k-1), m is the number of parts), subtract 1 from p(m), 2 from p(m-1), 4 from p(m-2), etc. (this gives a valid partition of the same type), add the part 2^m-1 to the other (initially empty) partition P', repeat until P is empty. The other direction goes by splitting parts 2^k-1 (uniquely) into distinct powers of 2 that are (in decreasing order) added at the left. - Joerg Arndt, Jan 06 2013

REFERENCES

Steenrod, N. and Epstein, D., Cohomology Operations, Princeton Univ. Press, 1962.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 513 terms from Reinhard Zumkeller)

P. C. P. Bhatt, An interesting way to partition a number, Inform. Process. Lett., 71, 1999, 141-148.

W. M. Y. Goh, P. Hitczenko and A. Shokoufandeh, s-partitions, Inform. Process. Lett., 82, 2002, 327-329.

Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

FORMULA

G.f.: 1/Product_{i>=1} (1 - x^(2^i-1)). - Simon Plouffe (corrected by Joerg Arndt, Dec 28 2012)

a(n) = p(n,1) with p(n,k) = if k <= n then p(n-k,k) + p(n,2*k+1), otherwise 0^n. - Reinhard Zumkeller, Mar 18 2009

G.f.: Sum_{i>=0} x^(2^i-1) / Product_{j=1..i} (1 - x^(2^j-1)). - Ilya Gutkovskiy, Jun 05 2017

EXAMPLE

From Joerg Arndt, Dec 28 2012: (Start)

There are a(17)=13 partitions of 17 into Mersenne numbers:

[ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]

[ 2]  [ 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]

[ 3]  [ 3 3 1 1 1 1 1 1 1 1 1 1 1 ]

[ 4]  [ 3 3 3 1 1 1 1 1 1 1 1 ]

[ 5]  [ 3 3 3 3 1 1 1 1 1 ]

[ 6]  [ 3 3 3 3 3 1 1 ]

[ 7]  [ 7 1 1 1 1 1 1 1 1 1 1 ]

[ 8]  [ 7 3 1 1 1 1 1 1 1 ]

[ 9]  [ 7 3 3 1 1 1 1 ]

[10]  [ 7 3 3 3 1 ]

[11]  [ 7 7 1 1 1 ]

[12]  [ 7 7 3 ]

[13]  [ 15 1 1 ]

There are a(17)=13 partitions p(1)+p(2)+...+p(m) = 17 such that 2*p(k) <= p(k-1):

[ 1]  [ 10 4 2 1 ]

[ 2]  [ 10 5 2 ]

[ 3]  [ 11 4 2 ]

[ 4]  [ 11 5 1 ]

[ 5]  [ 12 4 1 ]

[ 6]  [ 12 5 ]

[ 7]  [ 13 3 1 ]

[ 8]  [ 13 4 ]

[ 9]  [ 14 2 1 ]

[10]  [ 14 3 ]

[11]  [ 15 2 ]

[12]  [ 16 1 ]

[13]  [ 17 ]

(End)

MAPLE

The sequence is C(n, n) where C := proc(m, n) option remember; local k, a; if m = 0 then if n = 0 then 1 else 0 fi; elif m > n then C(n, n); else a := 0; for k from 0 to m do a := a + C(floor(k/2), n-k) od; a; fi end;

g:=1/product(1-x^(2^k-1), k=1..10): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..64); # Emeric Deutsch, Mar 06 2006

# alternative Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1,

      add(b(n-j, min(n-j, iquo(j, 2))), j=1..i))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..80);  # Alois P. Heinz, Mar 14 2021

MATHEMATICA

nn = 63; CoefficientList[

Series[Product[1/(1 - x^(2^i - 1)), {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 09 2013 *)

PROG

(PARI)

N=166; q='q+O('q^N);

gf=1/prod(n=1, 1+ceil(log(N)/log(2)), 1-q^(2^n - 1) );

Vec(gf)

/* Joerg Arndt, Oct 06 2012 */

CROSSREFS

Cf. A000225, A000041, A018819, A079559, A117145.

Sequence in context: A289139 A094838 A025768 * A325358 A029146 A029053

Adjacent sequences:  A000926 A000927 A000928 * A000930 A000931 A000932

KEYWORD

nonn

AUTHOR

J. Daniel Christensen, Mar 15 1996

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)