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%I M1043 N0392 #130 Sep 08 2022 08:44:28
%S 1,2,4,7,11,16,23,31,41,53,67,83,102,123,147,174,204,237,274,314,358,
%T 406,458,514,575,640,710,785,865,950,1041,1137,1239,1347,1461,1581,
%U 1708,1841,1981,2128,2282,2443,2612,2788,2972,3164,3364,3572,3789,4014
%N Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)).
%C Molien series for 4-dimensional representation of S_3 [Nebe, Rains, Sloane, Chap. 7].
%C From _Thomas Wieder_, Feb 11 2007: (Start)
%C 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 L. Shepherd_, Feb 27 2004.
%C For example, a(n=6)=16 because there are 16 integer partitions of n=3,4,...,n+2=8 with k=3 parts:
%C [[1, 1, 1]],
%C [[2, 1, 1]],
%C [[3, 1, 1], [2, 2, 1]]
%C [[4, 1, 1], [3, 2, 1], [2, 2, 2]],
%C [[5, 1, 1], [4, 2, 1], [3, 3, 1], [3, 2, 2]],
%C [[6, 1, 1], [5, 2, 1], [4, 3, 1], [4, 2, 2], [3, 3, 2]]. (End)
%C 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_, Feb 20 2007
%C Number of equivalence classes of 3 X n binary matrices when one can permute rows, permute columns and complement columns. - _Max Alekseyev_, Feb 05 2010
%C Convolution of the sequences whose n-th terms are given by 1+[n/2] and 1+[n/3], where []=floor. - _Clark Kimberling_, May 28 2012
%C Number of partitions of n into two sorts of 1, and one sort each of 2 and 3. - _Joerg Arndt_, May 05 2014
%C a(n-3) is the number of partitions mu of 2n of length 4 such that mu has an even number of even entries and the transpose of mu has an even number of even entries (see below example). - _John M. Campbell_, Feb 03 2016
%C Number of partitions of 2n+8 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both odd. Also, number of partitions of 2n+4 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both even. - _Wesley Ivan Hurt_, Jan 19 2021
%D A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A000601/b000601.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A. Cayley, <a href="/A001971/a001971.pdf">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
%H L. Colmenarejo, <a href="http://arxiv.org/abs/1604.00803">Combinatorics on several families of Kronecker coefficients related to plane partitions</a>, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
%H Florent de Dinechin, Matei Istoan, Guillaume Sergent, Kinga Illyes, Bogdan Popa and Nicolas Brunie, <a href="https://hal.inria.fr/ensl-00738412">Arithmetic around the bit heap</a>, HAL Id: ensl-00738412, 2012. - From _N. J. A. Sloane_, Dec 31 2012
%H E. Fix and J. L. Hodges, Jr., <a href="http://www.jstor.org/stable/2236885">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312.
%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
%H M. A. Harrison, <a href="http://dx.doi.org/10.1109/T-C.1973.223649">On the number of classes of binary matrices</a>, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649 - _Max Alekseyev_, Feb 05 2010
%H H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01359a034">The number of isomeric hydrocarbons of the methane series</a>, J. Amer. Chem. Soc., 53 (1931), 3077-3085.
%H H. R. Henze and C. M. Blair, <a href="/A000601/a000601_1.pdf">The number of isomeric hydrocarbons of the methane series</a>, J. Amer. Chem. Soc., 53 (1931), 3077-3085. (Annotated scanned copy)
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=196">Encyclopedia of Combinatorial Structures 196</a>
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H 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).
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1)
%F a(n) = n^3/36 +7*n^2/24 +11*n/12 +119/144 +(-1)^n/16 + A057078(n)/9. - _R. J. Mathar_, Mar 14 2011
%F a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=11, a(5)=16, a(6)=23, a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). - _Harvey P. Dale_, Mar 17 2013
%F It appears that a(n) = ((4*n^3+42*n^2+140*n+102+21*(1+(-1)^n))/8-6*floor((2*n+5+3*(-1)^n)/12))/18. - _Luce ETIENNE_, May 05 2014
%F Euler transform of length 3 sequence [ 2, 1, 1]. - _Michael Somos_, May 28 2014
%F a(-7 - n) = -a(n). - _Michael Somos_, May 28 2014
%e G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 23*x^6 + 31*x^7 + ...
%e From _John M. Campbell_, Feb 03 2016: (Start)
%e For example, letting n=6, there are a(n-3)=a(3)=7 partitions mu of 12 of length 4 such mu has an even number of even entries and the transpose of mu has an even number of even entries: (8,2,1,1), (6,4,1,1), (6,3,2,1), (6,2,2,2), (4,4,3,1), (4,4,2,2), (4,3,3,2). For example, the partition
%e oooooo
%e ooo
%e oo
%e o
%e has 2 even entries and the transpose
%e oooo
%e ooo
%e oo
%e o
%e o
%e o
%e has an even number of even entries. (End)
%p A000601:=1/(z+1)/(z**2+z+1)/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%p 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_, Feb 07 2008
%t CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)), {x, 0, 49}], x] (* _Jean-François Alcover_, Jul 20 2011 *)
%t LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,4,7,11,16,23},50] (* _Harvey P. Dale_, Mar 17 2013 *)
%t a[ n_] := Quotient[ 2 n^3 + 21 n^2 + 66 n, 72] + 1; (* _Michael Somos_, May 28 2014 *)
%o (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);
%o (PARI) Vec(1/((1-x)^2*(1-x^2)*(1-x^3))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (PARI) {a(n) = (2*n^3 + 21*n^2 + 66*n) \ 72 + 1}; /* _Michael Somos_, May 28 2014 */
%Y Cf. A002620, A006148, A006383.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Feb 06 2000
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