A265055 Expansion of b(2)*b(4)*b(6)/(x^8-x^4-x+1), where b(k) = (1-x^k)/(1-x).

1, 4, 9, 16, 25, 37, 53, 74, 101, 135, 179, 237, 313, 411, 537, 700, 912, 1188, 1546, 2009, 2608, 3385, 4394, 5703, 7399, 9596, 12444, 16138, 20929, 27140, 35190, 45625, 59155, 76699, 99445, 128932, 167158, 216717, 280972, 364279, 472282, 612300, 793827, 1029174, 1334298

Offset

0,2

Comments

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_1 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 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.

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 17.supp01 (2010): 169-215.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,0,0,0,-1).

Formula

G.f.: (1 + x^2)*(1 - x + x^2)*(1 + x + x^2)*(1 + x)^3/((1 - x)*(1 - x^4 - x^5 - x^6 - x^7)). [Bruno Berselli, Dec 28 2015]

Mathematica

Join[{1, 4}, LinearRecurrence[{1, 0, 0, 1, 0, 0, 0, -1}, {9, 16, 25, 37, 53, 74, 101, 135}, 50]] (* Jean-François Alcover, Jan 08 2019 *)

Prog

(Magma) /* By definition: */ m:=50; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)*b(4)*b(6)/(x^8-x^4-x+1))); // Bruno Berselli, Dec 28 2015

Crossrefs

For diagrams QL4_1 - QL4_22 see: A265055, A266333, A266334, A266335, A266336, A266338, A266339, A266340, A266370, A266371, A266372, A266373, A266374, A266375, A266376, A266367, A266353, A266355, A266354, A264079, A266337, A162740 respectively.

Sequence in context: A008088 A181640 A033616 * A011895 A033612 A033610

Adjacent sequences: A265052 A265053 A265054 * A265056 A265057 A265058

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved