A266367 Expansion of b(2)*b(4)/(1 - 2*x - 2*x^3 + 3*x^4), where b(k) = (1-x^k)/(1-x).

1, 4, 10, 24, 54, 116, 250, 536, 1142, 2436, 5194, 11064, 23574, 50228, 107002, 227960, 485654, 1034628, 2204170, 4695768, 10003830, 21312116, 45403258, 96726872, 206066486, 439003140, 935250250, 1992452856, 4244712534, 9042916148, 19264987258, 41042041016, 87435776726

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_16 - 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 (2,0,2,-3)

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)/(1-2*x-2*x^3+3*x^4)));

Mathematica

CoefficientList[Series[(1 + x)^2 (1 + x^2)/((1 - x) (1 - x - x^2 - 3 x^3)), {x, 0, 40}], x]
LinearRecurrence[{2,0,2,-3},{1,4,10,24,54},40] (* Harvey P. Dale, Mar 22 2016 *)

Formula

G.f.: (1 + x)^2*(1 + x^2)/((1 - x)*(1 - x - x^2 - 3*x^3)).

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