A264079 Expansion of b(2)*b(6)/(1 - 2*x + x^3 - x^4 - x^5 + 2*x^6), where b(k) = (1-x^k)/(1-x).

1, 4, 10, 21, 41, 79, 150, 282, 527, 982, 1829, 3405, 6337, 11790, 21932, 40797, 75888, 141161, 262573, 488407, 908474, 1689830, 3143211, 5846606, 10875117, 20228513, 37626513, 69988066, 130182920, 242149745, 450416216, 837807065, 1558382345, 2898705007, 5391803070

Offset: 0

Author

Bruno Berselli, Dec 28 2015

Keywords

• nonn
• easy

Comments

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_20 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009, page 31.
• Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics, Volume 17, Supplement 1 (2010), page 186.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1,1,-2).

Crossrefs

Cf. similar sequences listed in A265055.

Programs

Magma

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(6)/(1-2*x+x^3-x^4-x^5+2*x^6)));

Mathematica

CoefficientList[Series[(1 + x)^2 (1 - x + x^2) (1 + x + x^2)/((1 - x) (1 - x - x^2 - x^4 - 2 x^5)), {x, 0, 40}], x]
LinearRecurrence[{2,0,-1,1,1,-2},{1,4,10,21,41,79,150},40] (* Harvey P. Dale, Jan 18 2021 *)

Formula

G.f.: (1 + x)^2*(1 - x + x^2)*(1 + x + x^2)/((1 - x)*(1 - x - x^2 - x^4 - 2*x^5)).

a(n) = 2*a(n-1) - a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) for n>6.