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%I M2148 #45 Jul 10 2023 02:29:36
%S 0,2,28,168,660,2002,5096,11424,23256,43890,77924,131560,212940,
%T 332514,503440,742016,1068144,1505826,2083692,2835560,3801028,5026098,
%U 6563832,8475040,10829000,13704210,17189172,21383208,26397308,32355010,39393312,47663616,57332704
%N From the enumeration of corners.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A006332/b006332.txt">Table of n, a(n) for n = 0..1000</a>
%H Germain Kreweras, <a href="http://www.numdam.org/numdam-bin/item?id=BURO_1965__6__9_0">Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle}, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
%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 (7,-21,35,-35,21,-7,1).
%F a(n) = (n*(1 + n)^2*(2 + n)*(1 + 2*n)*(3 + 2*n))/90.
%F a(n) = 2*A006858(n).
%F a(n) = (-1)^(n+1)*A132339(3, n).
%F G.f.: 2*(1+x)*(1 + 6*x + x^2)/(1-x)^7.
%F From _G. C. Greubel_, Dec 14 2021: (Start)
%F E.g.f.: (1/90)*x*(180 + 1080*x + 1350*x^2 + 555*x^3 + 84*x^4 + 4*x^5)*exp(x).
%F a(n) = binomial(n+2, 3)*binomial(2*n+3, 3)/5. (End)
%F From _Amiram Eldar_, Jul 10 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 15*Pi^2 - 295/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2/2 + 120*Pi - 605/2. (End)
%p A006332:=-2*(1+z)*(z**2+6*z+1)/(z-1)**7; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[(n(1+n)^2(2+n)(1+2n)(3+2n))/90, {n, 0, 30}] (* or *)
%t {0}~Join~CoefficientList[Series[2(x+1)(x^2 +6x +1)/(1-x)^7, {x, 0, 29}], x] (* _Michael De Vlieger_, Mar 26 2016 *)
%o (PARI) my(x='x+O('x^99)); concat(0, Vec(2*(x+1)*(x^2+6*x+1)/(1-x)^7)) \\ _Altug Alkan_, Mar 26 2016
%o (Magma) [Binomial(n+2, 3)*Binomial(2*n+3, 3)/5: n in [0..30]]; // _G. C. Greubel_, Dec 14 2021
%o (Sage) [binomial(n+2, 3)*binomial(2*n+3, 3)/5 for n in (0..30)] # _G. C. Greubel_, Dec 14 2021
%Y Cf. A006858, A132339.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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