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%I M2441 N0969 #56 Mar 30 2023 14:19:23
%S 1,1,3,5,8,12,18,24,33,43,55,69,86,104,126,150,177,207,241,277,318,
%T 362,410,462,519,579,645,715,790,870,956,1046,1143,1245,1353,1467,
%U 1588,1714,1848,1988
%N Expansion of (1+x^3)/((1-x)*(1-x^2)^2*(1-x^3)).
%C a(1..3)=0; a(n) is the number of partitions of 2*(n+1) with 4 different numbers from the set {1,...,n}; the number of partitions of 2*n + 2 - C and 2*n + 2 + C are equal; example: n=6; 2*n + 2 = 14; a(6)=3; (10,1), (11,1), (12,2), (13,2), (14,3), (15,2), (16,2), (17,1), (18,1). - _Paul Weisenhorn_, Jun 01 2009. [I believe this comment refers to the sequence 0, 0, 0, 1, 1, 3, 5, ... with offset 1. - _N. J. A. Sloane_, Mar 30 2023]
%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 M. Jeger, Einfuehrung in die Kombinatorik, Klett, 1975, pages 110ff. [From _Paul Weisenhorn_, Jun 01 2009]
%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="/A001973/b001973.txt">Table of n, a(n) for n = 0..1000</a>
%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 Shalosh B. Ekhad, Doron Zeilberger, <a href="https://arxiv.org/abs/1901.08172">In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?</a>, arXiv:1901.08172 [math.CO], 2019.
%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_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1)
%F a(n) is the coefficient of x^(2*n+2) in the g.f. Product_{s=1..4} (x^s - x^(n+1))/(1-x^s). - _Paul Weisenhorn_, Jun 01 2009
%F a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). _Vincenzo Librandi_, Jun 11 2012
%p A001973:=(1-z+z**2)/(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>=2)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=6..45) ; # _Zerinvary Lajos_, Feb 07 2008
%t CoefficientList[Series[(1+x^3)/((1-x)*(1-x^2)^2*(1-x^3)),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 11 2012 *)
%o (PARI) Vec((1+x^3)/((1-x)*(1-x^2)^2*(1-x^3))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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