A266373 G.f. = b(2)*b(6)*b(10)/(x^15+x^14+x^13+x^12+x^11-2*x^5-x^4-x^3-x^2-x+1), where b(k) = (1-x^k)/(1-x).

1, 4, 10, 22, 46, 95, 193, 388, 778, 1558, 3118, 6236, 12468, 24926, 49830, 99614, 199133, 398073, 795758, 1590738, 3179918, 6356718, 12707200, 25401932, 50778938, 101508046, 202916478, 405633823, 810869573, 1620943388, 3240296038, 6477412158, 12948467598

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_12 - 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, -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)/(x^15+x^14+x^13 +x^12+x^11-2*x^5-x^4-x^3-x^2-x+1))); // Bruno Berselli, Dec 29 2015

Maple

gf:= b(2)*b(6)*b(10)/(x^15+x^14+x^13+x^12+x^11-2*x^5-x^4-x^3-x^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[2] b[6] b[10]/(x^15 + x^14 + x^13 + x^12 + x^11 - 2 x^5 - x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 28 2015 *)
LinearRecurrence[{2,0,0,0,0,0,0,0,0,0,-1},{1,4,10,22,46,95,193,388,778,1558,3118,6236},40] (* Harvey P. Dale, Mar 14 2016 *)