Search: id:A000601
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%I A000601 M1043 N0392
%S A000601 1,2,4,7,11,16,23,31,41,53,67,83,102,123,147,174,204,237,274,314,358,
%T A000601 406,458,514,575,640,710,785,865,950,1041,1137,1239,1347,1461,1581,
%U A000601 1708,1841,1981,2128,2282,2443,2612,2788,2972,3164,3364,3572,3789,4014
%N A000601 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)).
%C A000601 Molien series for 4-dimensional representation of S_3 [Nebe, Rains, Sloane, 
               Chap. 7].
%C A000601 Comments from Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 11 2007: 
               (Start) If P(i,k) denotes the number of integer partitions of i into 
               k parts and if k=3, then a(n)=sum_{i=k}^{n+2} P(i,k). See also A002620 
               = Quarter-squares, this sequence follows for k=2 as pointed out by 
               Rick Shepherd (rshepherd2(AT)hotmail.com), Feb 27 2004.
%C A000601 For example, a(n=6)=16 because there are 16 integer partitions of n=3,
               4,...,n+2=8 with k=3 parts:
%C A000601 [[1, 1, 1]],
%C A000601 [[2, 1, 1]],
%C A000601 [[3, 1, 1], [2, 2, 1]]
%C A000601 [[4, 1, 1], [3, 2, 1], [2, 2, 2]],
%C A000601 [[5, 1, 1], [4, 2, 1], [3, 3, 1], [3, 2, 2]],
%C A000601 [[6, 1, 1], [5, 2, 1], [4, 3, 1], [4, 2, 2], [3, 3, 2]]. (End)
%C A000601 Let P(i,k) be the number of integer partitions of n into k parts. Then 
               if k=3 we have a(n)=sum_{i=k}^{n} P(i,k=3). - Thomas Wieder (thomas.wieder(AT)t-online.de), 
               Feb 20 2007
%D A000601 A. Cayley, Numerical tables supplementary to second memoir on quantics, 
               Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, 
               London, 1889-1897, Vol. 2, p. 276-281.
%D A000601 E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon 
               test, Annals Math. Stat., 26 (1955), 301-312.
%D A000601 H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the 
               methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.
%D A000601 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000601 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000601 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000601 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=196">
               Encyclopedia of Combinatorial Structures 196</a>
%H A000601 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/
               ~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, 
               Springer, Berlin, 2006.
%H A000601 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000601 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000601 Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http:/
               /www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</
               a>, vol. 8 (2008).
%p A000601 A000601:=1/(z+1)/(z**2+z+1)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%p A000601 with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), 
               right=Set(U, card<r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=2, 
               stack): seq(count(subs(r=2, ZL), size=m), m=3..52) ; - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2008
%o A000601 (MAGMA) K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,
               -1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,
               [1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; U:=MatrixGroup<4,
               K|q2,h>; G:=MatrixGroup<4,K|q1,q2,h>; H:=MatrixGroup<4,K|q1,q2,h,
               p1>; MolienSeries(U);
%Y A000601 Cf. A002620.
%Y A000601 Sequence in context: A131075 A133523 A114805 this_sequence A062433 A065095 
               A005253
%Y A000601 Adjacent sequences: A000598 A000599 A000600 this_sequence A000602 A000603 
               A000604
%K A000601 nonn
%O A000601 0,2
%A A000601 N. J. A. Sloane (njas(AT)research.att.com).
%E A000601 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 06 2000

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