A096338 a(n) = (2/(n-1))*a(n-1) + ((n+5)/(n-1))*a(n-2) with a(0)=0 and a(1)=1.

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

Offset: 0

Author

Benoit Cloitre, Jun 28 2004

Keywords

• nonn
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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.
• Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 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.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A006918, A038163, A060099, A058187, A000292, A028724.

Programs

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 *)

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