A266354 Expansion of b(2)*b(6)*b(10)/(1 - x - x^2 - x^4 - x^5 + x^11 + x^12 + x^14), where b(k) = (1-x^k)/(1-x).

1, 4, 10, 21, 41, 78, 145, 266, 485, 882, 1601, 2902, 5256, 9516, 17226, 31180, 56435, 102143, 184868, 334588, 605559, 1095976, 1983558, 3589950, 6497282, 11759123, 21282277, 38517777, 69711482, 126167473, 228344464, 413269701, 747957021, 1353691555, 2449981446

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_19 - 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,-1,1,0,-1,1,-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(6)*b(10)/(1-x-x^2-x^4-x^5+x^11+x^12+x^14)));

Mathematica

CoefficientList[Series[(1 + x)^3 (1 - x + x^2) (1 + x + x^2) (1 - x + x^2 - x^3 + x^4)/((1 - x) (1 - x - x^2 - x^4 - x^6 - x^7 - x^9)), {x, 0, 40}], x]

Formula

G.f.: (1 + x)^3*(1 - x + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)/((1 - x)*(1 - x - x^2 - x^4 - x^6 - x^7 - x^9)).