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 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 , [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 ]   [ 7 3 3 3 1 ]   [ 7 7 1 1 1 ]   [ 7 7 3 ]   [ 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 ]   [ 14 3 ]   [ 15 2 ]   [ 16 1 ]   [ 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

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