%I A000726 M0316 N0116
%S A000726 1,1,2,2,4,5,7,9,13,16,22,27,36,44,57,70,89,108,135,163,202,243,297,355,
%T A000726 431,513,617,731,874,1031,1225,1439,1701,1991,2341,2731,3197,3717,4333,
%U A000726 5022,5834,6741,7803,8991,10375,11923,13716,15723,18038,20628,23603
%N A000726 Number of partitions of n in which no parts are multiples of 3.
%C A000726 Case k=4, i=3 of Gordon Theorem.
%C A000726 Expansion of q^(-1/12)eta(q^3)/eta(q) in powers of q. - Michael Somos 
               Apr 20 2004
%C A000726 Euler transform of period 3 sequence [1,1,0,...]. - Michael Somos Apr 
               20 2004
%C A000726 Also number of partitions with at most 2 parts of size 1 and all differences 
               between parts at distance 3 are greater than 1. Example: a(6)=7 because 
               we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,2] (for example, 
               [2,2,1,1] does not qualify because the difference between the first 
               and the fourth parts is equal to 1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 18 2006
%C A000726 Also number of partitions of n where no positive integer appears more 
               than twice. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,
               1],[3,3],[3,2,1] and [2,2,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 18 2006
%C A000726 Also number of partitions of n with least part either 1 or 2 and with 
               differences of consecutive parts at most 2. Example: a(6)=7 because 
               we have [4,2],[3,2,1],[3,1,1,1],[2,2,2],[2,2,1,1],[2,1,1,1,1] and 
               [1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 
               2006
%D A000726 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000726 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000726 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
%D A000726 L. Carlitz, Generating functions and partition problems, pp. 144-169 
               of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 
               8 (1965). Amer. Math. Soc., see p. 145.
%H A000726 T. D. Noe, <a href="b000726.txt">Table of n, a(n) for n=0..1000</a>
%H A000726 N. Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities 
               from Partial Supersymmetry</a>
%H A000726 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionb.html">Link to a section of The World of Mathematics.</
               a>
%F A000726 Given g.f. A(x) then B(x)=x*A(x^6)^2 satisfies 0=f(B(x), B(x^2), B(x^4)) 
               where f(u,v,w)= +v^2 +v*w^2 -v*u^2 +3*u^2*w^2 . - Michael Somos May 
               28 2006
%F A000726 G.f.: 1/(Product_{k>0} (1-x^(3k-1))(1-x^(3k-2))) = Product_{k>0} 1+x^k+x^(2k) 
               (where 1+x+x^2 is 3rd cyclotomic polynomial).
%e A000726 a(6)=7 because we have [5,1],[4,2],[4,1,1],[2,2,2],[2,2,1,1],[2,1,1,1,
               1] and [1,1,1,1,1,1].
%p A000726 g:=product(1+x^j+x^(2*j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,
               x,n),n=0..50); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 
               2006
%t A000726 f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, 
               n}], x^n]; Table[f@n, {n, 0, 40}] (* from Robert G. Wilson v (rgwv(at)rgwv.com), 
               Nov 10 2006 *)
%o A000726 (PARI) a(n)=if(n<0,0,polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)),n))
%Y A000726 Cf. A001935, A035959. a(n)=A061197(n, n).
%Y A000726 Cf. A001651, A003105, A035361, A035360.
%Y A000726 Sequence in context: A166239 A058661 A094362 this_sequence A128663 A135833 
               A137200
%Y A000726 Adjacent sequences: A000723 A000724 A000725 this_sequence A000727 A000728 
               A000729
%K A000726 nonn,easy,nice
%O A000726 0,3
%A A000726 N. J. A. Sloane (njas(AT)research.att.com).
%E A000726 More terms from Olivier Gerard (olivier.gerard(AT)gmail.com)