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Expansion of 1/((1-x)^3*(1-x^2)^2*(1-x^3)).
(Formerly M2726 N1093)
5

%I M2726 N1093 #54 Sep 24 2022 13:44:32

%S 1,3,8,17,33,58,97,153,233,342,489,681,930,1245,1641,2130,2730,3456,

%T 4330,5370,6602,8048,9738,11698,13963,16563,19538,22923,26763,31098,

%U 35979,41451,47571,54390,61971,70371,79660,89901,101171,113540,127092,141904,158068,175668,194804,215568

%N Expansion of 1/((1-x)^3*(1-x^2)^2*(1-x^3)).

%C Number of (integer) partitions of n into 3 sorts of 1's, 2 sorts of 2's, and 1 sort of 3's. - _Joerg Arndt_, May 17 2013

%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 Vincenzo Librandi, <a href="/A002625/b002625.txt">Table of n, a(n) for n = 0..1000</a>

%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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=205">Encyclopedia of Combinatorial Structures 205</a>

%H Gerzson Keri and Patric R. J. Östergård, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Keri/keri6.html">The Number of Inequivalent (2R+3,7)R Optimal Covering Codes</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.

%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 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-4,2,2,2,-4,-1,3,-1).

%F a(n) = floor((n+1)*(135*(-1)^n + 6*n^4 + 144*n^3 + 1256*n^2 + 4744*n + 6785)/8640+1/2). - _Tani Akinari_, Oct 07 2012

%p A002625:=1/(z**2+z+1)/(z+1)**2/(z-1)**6; [_Simon Plouffe_ in his 1992 dissertation.]

%t CoefficientList[Series[1/((1-x)^3*(1-x^2)^2*(1-x^3)),{x,0,50}],x] (* _Vincenzo Librandi_, Feb 25 2012 *)

%t LinearRecurrence[{3,-1,-4,2,2,2,-4,-1,3,-1},{1,3,8,17,33,58,97,153,233,342},50] (* _Harvey P. Dale_, Sep 24 2022 *)

%o (PARI) Vec(1/(1-x)^3/(1-x^2)^2/(1-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Apr 30 2012

%Y Partial sums of A097701.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_