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A000929 Dimension of n-th degree part of Steenrod algebra. 9
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

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.

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

LINKS

R. Zumkeller, Table of n, a(n) for n = 0..512

FORMULA

G.f.: 1/prod(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) else 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

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 * A029146 A029053 A053254

Adjacent sequences:  A000926 A000927 A000928 * A000930 A000931 A000932

KEYWORD

nonn

AUTHOR

J. Daniel Christensen, Mar 15 1996

STATUS

approved

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Last modified August 19 07:12 EDT 2017. Contains 290794 sequences.