A266371 G.f. = b(4)*b(6)/(x^8+x^6-x^5-x^3+x^2-2*x+1), where b(k) = (1-x^k)/(1-x).

1, 4, 10, 21, 40, 74, 135, 244, 438, 782, 1394, 2484, 4425, 7880, 14028, 24969, 44442, 79102, 140792, 250588, 446002, 793801, 1412820, 2514562, 4475459, 7965488, 14177086, 25232574, 44909290, 79930188, 142260869, 253197876, 450645100, 802064421, 1427525430

Offset: 0

Author

Alois P. Heinz, Dec 28 2015

Keywords

• nonn
• easy

Comments

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_10 - 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 (2,-1,1,0,1,-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(4)*b(6)/(x^8+x^6-x^5-x^3+x^2-2*x+1))); // Bruno Berselli, Dec 29 2015

Maple

gf:= b(4)*b(6)/(x^8+x^6-x^5-x^3+x^2-2*x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

Mathematica

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[4] b[6]/(x^8 + x^6 - x^5 - x^3 + x^2 - 2 x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 28 2015 *)
LinearRecurrence[{2,-1,1,0,1,-1,0,-1},{1,4,10,21,40,74,135,244,438},40] (* Harvey P. Dale, Nov 06 2017 *)