%I A001935 M0566 N0204
%S A001935 1,1,2,3,4,6,9,12,16,22,29,38,50,64,82,105,132,166,208,258,320,395,484,
%T A001935 592,722,876,1060,1280,1539,1846,2210,2636,3138,3728,4416,5222,6163,
%U A001935 7256,8528,10006,11716,13696,15986,18624,21666,25169,29190,33808,39104
%N A001935 Number of partitions with no even part repeated; partitions of n in which 
               no parts are multiples of 4
%C A001935 Also number of partitions of n where no part appears more than three 
               times.
%C A001935 Euler transform of period 4 sequence [1,1,1,0,...].
%C A001935 Expansion of q^(-1/8)eta(q^4)/eta(q) in powers of q. - Michael Somos 
               Mar 19 2004
%C A001935 a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n 
               is in A062717 (see Fink et al).
%C A001935 Also number of partitions of n in which the least part and the differences 
               between consecutive parts is at most 3. Example: a(5)=6 because we 
               have [4,1],[3,2],[3,1,1],[2,2,1],[2,1,1,1] and [1,1,1,1,1]. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Apr 19 2006
%D A001935 G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 
               44 (No. 4, 2007), 561-573. (See Th. 9.)
%D A001935 A. Cayley, A memoir on the transformation of elliptic functions, Collected 
               Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, 
               Vol. 9, p. 128.
%D A001935 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring 
               at most thrice, in preparation.
%D A001935 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
%D A001935 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001935 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001935 T. D. Noe, <a href="b001935.txt">Table of n, a(n) for n=0..1000</a>
%H A001935 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionb.html">Link to a section of The World of Mathematics.</
               a>
%H A001935 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PartitionFunctionP.html">Partition Function P</a>
%F A001935 G.f.: Product(j=1, oo, 1 + x^j + x^2j + x^3j) - Jon Perry (perry(AT)globalnet.co.uk), 
               Mar 30 2004
%F A001935 G.f.: product(k=1, oo, (1+x^k)^(2-k%2)) - Jon Perry (perry(AT)globalnet.co.uk), 
               May 05 2005
%F A001935 G.f.: Product_{k>0} (1+x^(2k))/(1-x^(2k-1)) = 1+Sum_{k>0}(Product_{i=1..k} 
               (x^i+1)/(x^-i-1)).
%e A001935 a(5)=6 because we have [5],[4,1],[3,2],[3,1,1],[2,1,1,1] and [1,1,1,1,
               1].
%p A001935 g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,
               x,n),n=0..48); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 19 
               2006
%o A001935 (PARI) a(n)=if(n<0,0,polcoeff(eta(x^4+x*O(x^n))/eta(x+x*O(x^n)),n))
%o A001935 (PARI) a(n)=if(n<0,0,polcoeff(sum(k=0,(sqrtint(8*n+1)-1)\2,prod(i=1,k,
               (1+x^i)/(x^-i-1),1+x*O(x^n))),n)) /* Michael Somos Jun 01 2004 */
%Y A001935 A083365(n)=(-1)^n a(n). Convolution square is A001936. Cf. A000009, A000726, 
               A035959, A061198, A061199.
%Y A001935 Equals A098491 + A098492.
%Y A001935 Sequence in context: A058647 A073576 A069907 this_sequence A083365 A007604 
               A013950
%Y A001935 Adjacent sequences: A001932 A001933 A001934 this_sequence A001936 A001937 
               A001938
%K A001935 nonn,easy,nice
%O A001935 0,3
%A A001935 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Robert G. 
               Wilson v (rgwv(AT)rgwv.com)
%E A001935 More terms from James A. Sellers (sellersj(AT)math.psu.edu)