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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 Steenrod, N. and Epstein, D., Cohomology Operations, Princeton Univ. Press, 1962. LINKS R. Zumkeller, Table of n, a(n) for n = 0..512 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/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 October 17 01:37 EDT 2018. Contains 316275 sequences. (Running on oeis4.)