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A096338 a(n) = (2/(n-1))*a(n-1) + ((n+5)/(n-1))*a(n-2) with a(0)=0 and a(1)=1. 11
0, 1, 2, 6, 10, 20, 30, 50, 70, 105, 140, 196, 252, 336, 420, 540, 660, 825, 990, 1210, 1430, 1716, 2002, 2366, 2730, 3185, 3640, 4200, 4760, 5440, 6120, 6936, 7752, 8721, 9690, 10830, 11970, 13300, 14630, 16170, 17710, 19481, 21252, 23276, 25300, 27600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Without the leading zero, Poincaré series [or Poincare series] P(C_{2,2}; t).
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
Starting with 1 = [1, 2, 3, ...] convolved with the aerated triangular series, [1, 0, 3, 0, 6, ...]. - Gary W. Adamson, Jun 11 2009
From Alford Arnold, Oct 14 2009: (Start)
a(n) is also related to Dyck Paths. Note that
0 1 2 6 10 20 30 50 70 105 ...
minus
0 0 0 0 1 2 6 10 20 30 ...
equals
0 1 2 6 9 18 24 40 50 75 ... A028724
(End)
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
LINKS
Dragomir Z. Djokovic, Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 3.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4, 19.
M. Navascues and T. Vertesi, The Structure of Matrix Product States, arXiv preprint arXiv:1509.04507 [quant-ph], 2015-2018.
Miguel Navascués and Tamás Vértesi, Bond dimension witnesses and the structure of homogeneous matrix product states, Quantum 2 (2018): 50.
FORMULA
G.f.: x/((1-x)^2*(1-x^2)^3). - Ralf Stephan, Jun 29 2004
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
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
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
MAPLE
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
MATHEMATICA
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 *)
CoefficientList[Series[x/((1 - x)^2*(1 - x^2)^3), {x, 0, 45}], x] (* or *)
Nest[Append[#1, (2/(#2 - 1))*#1[[-1]] + ((#2 + 5)/(#2 - 1))*#1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 44] (* Michael De Vlieger, May 30 2018 *)
CROSSREFS
Sequence in context: A320942 A168152 A211982 * A198381 A309846 A309874
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Jun 28 2004
STATUS
approved

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Last modified May 20 23:09 EDT 2024. Contains 372720 sequences. (Running on oeis4.)