A266353 - OEIS

A266353

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

1, 4, 10, 20, 35, 57, 89, 136, 205, 306, 454, 671, 989, 1455, 2138, 3139, 4606, 6756, 9907, 14525, 21293, 31212, 45749, 67054, 98278, 144039, 211105, 309395, 453446, 664563, 973970, 1427428, 2092003, 3065985, 4493425, 6585440, 9651437, 14144874, 20730326, 30381775

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_17 - 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.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

FORMULA

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

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

a(n) = a(n-1) + a(n-3) + 12 for n>4. - Greg Dresden, Feb 09 2020

MATHEMATICA

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

PROG

(Magma) /* By definition: */ m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(3)*b(4)/(1-2*x+x^2-x^3+x^4)));