A266355 Expansion of b(2)*b(4)*b(6)/(1-x-x^2-x^4+x^6+x^8), where b(k) = (1-x^k)/(1-x).

1, 4, 10, 21, 40, 73, 129, 224, 385, 658, 1122, 1910, 3248, 5519, 9375, 15922, 27038, 45911, 77954, 132358, 224727, 381555, 647823, 1099903, 1867461, 3170650, 5383253, 9139893, 15518057, 26347142, 44733168, 75949650, 128950161, 218936410, 371718429, 631117454

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_18 - 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 (1,1,0,1,0,-1,0,-1).

Crossrefs

Cf. similar sequences listed in A265055.

Programs

Magma

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(4)*b(6)/(1-x-x^2-x^4+x^6+x^8)));

Mathematica

CoefficientList[Series[(1 + x^2) (1 - x + x^2) (1 + x + x^2) (1 + x)^3/((1 - x) (1 - x^2 - x^3 - 2 x^4 - 2 x^5 - x^6 - x^7)), {x, 0, 40}], x]
LinearRecurrence[{1,1,0,1,0,-1,0,-1},{1,4,10,21,40,73,129,224,385,658},40] (* Harvey P. Dale, Apr 04 2019 *)

Formula

G.f.: (1 + x^2)*(1 - x + x^2)*(1 + x + x^2)*(1 + x)^3/((1 - x)*(1 - x^2 - x^3 - 2*x^4 - 2*x^5 - x^6 - x^7)).

a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6) - a(n-8) for n>9.