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%I #68 May 05 2023 12:16:57
%S 0,1,2,6,10,20,30,50,70,105,140,196,252,336,420,540,660,825,990,1210,
%T 1430,1716,2002,2366,2730,3185,3640,4200,4760,5440,6120,6936,7752,
%U 8721,9690,10830,11970,13300,14630,16170,17710,19481,21252,23276,25300,27600
%N a(n) = (2/(n-1))*a(n-1) + ((n+5)/(n-1))*a(n-2) with a(0)=0 and a(1)=1.
%C Without the leading zero, Poincaré series [or Poincare series] P(C_{2,2}; t).
%C Starting (1, 2, 6, ...) = partial sums of the tetrahedral numbers, A000292 with repeats: (1, 1, 4, 4, 10, 10, 20, 20, 35, 35, ...). - _Gary W. Adamson_, Mar 30 2009
%C Starting with 1 = [1, 2, 3, ...] convolved with the aerated triangular series, [1, 0, 3, 0, 6, ...]. - _Gary W. Adamson_, Jun 11 2009
%C From _Alford Arnold_, Oct 14 2009: (Start)
%C a(n) is also related to Dyck Paths. Note that
%C 0 1 2 6 10 20 30 50 70 105 ...
%C minus
%C 0 0 0 0 1 2 6 10 20 30 ...
%C equals
%C 0 1 2 6 9 18 24 40 50 75 ... A028724
%C (End)
%C The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums of A139600 are related to the sequence given above; e.g., Ze1(n) = 3*A096338(n-1) - 3*A096338(n-3) + 9*A096338(n-4), with A096338(n) = 0 for n <= -1. For the definition of these triangle sums, see A180662. - _Johannes W. Meijer_, Apr 29 2011
%H Vincenzo Librandi, <a href="/A096338/b096338.txt">Table of n, a(n) for n = 0..1000</a>
%H Dragomir Z. Djokovic, <a href="https://arxiv.org/abs/math/0609262">Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices</a>, arXiv:math/0609262 [math.AC], 2006. See Table 3.
%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4, 19.
%H M. Navascues and T. Vertesi, <a href="https://arxiv.org/abs/1509.04507">The Structure of Matrix Product States</a>, arXiv preprint arXiv:1509.04507 [quant-ph], 2015-2018.
%H Miguel Navascués and Tamás Vértesi, <a href="https://doi.org/10.22331/q-2018-01-31-50">Bond dimension witnesses and the structure of homogeneous matrix product states</a>, Quantum 2 (2018): 50.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F G.f.: x/((1-x)^2*(1-x^2)^3). - _Ralf Stephan_, Jun 29 2004
%F a(n) = Sum_{k=1..floor(n/2)+1} ( Sum_{i=1..k} i*(n-2*k+2) ) = -(floor(n/2)+1) * (floor(n/2)+2) * (floor(n/2)+3) * (3*floor(n/2) - 2*n)/12. - _Wesley Ivan Hurt_, Sep 26 2013
%F a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8). - _Wesley Ivan Hurt_, Nov 26 2020
%F 128*a(n) = 8*n^3 +94/3*n^2 +44*n +15 +2/3*n^4 -2*(-1)^n*n^2 -12*(-1)^n*n -15*(-1)^n. - _R. J. Mathar_, Mar 23 2021
%p A096338:=n->-(floor(n/2)+1)*(floor(n/2)+2)*(floor(n/2)+3)*(3*floor(n/2)-2*n)/12; seq(A096338(k),k=0..100); # _Wesley Ivan Hurt_, Oct 04 2013
%t t = {0, 1}; Do[AppendTo[t, (2/(n - 1))*t[[-1]] + ((n + 5)/(n - 1))*t[[-2]]], {n, 2, 50}]; t (* _T. D. Noe_, Oct 08 2013 *)
%t CoefficientList[Series[x/((1 - x)^2*(1 - x^2)^3), {x, 0, 45}], x] (* or *)
%t Nest[Append[#1, (2/(#2 - 1))*#1[[-1]] + ((#2 + 5)/(#2 - 1))*#1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 44] (* _Michael De Vlieger_, May 30 2018 *)
%Y Cf. A006918, A038163, A060099, A058187, A000292, A028724.
%K nonn,easy
%O 0,3
%A _Benoit Cloitre_, Jun 28 2004
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