A266335 G.f. = b(2)^2*b(6)/(x^7+x^6-x^5-x^2-x+1), where b(k) = (1-x^k)/(1-x).

1, 4, 9, 17, 30, 52, 88, 145, 237, 386, 628, 1020, 1653, 2677, 4334, 7016, 11356, 18377, 29737, 48118, 77860, 125984, 203849, 329837, 533690, 863532, 1397228, 2260765, 3657997, 5918766, 9576768, 15495540, 25072313, 40567857, 65640174, 106208036, 171848216

Offset: 0

Author

Alois P. Heinz, Dec 27 2015

Keywords

• nonn
• easy

Comments

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_4 - 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 (1,1,0,0,1,-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)^2*b(6)/(x^7+x^6-x^5-x^2-x+1))); // Bruno Berselli, Dec 29 2015

Maple

gf:= b(2)^2*b(6)/(x^7+x^6-x^5-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]^2 b[6]/(x^7 + x^6 - x^5 - x^2 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 29 2015 *)
LinearRecurrence[{1,1,0,0,1,-1,-1},{1,4,9,17,30,52,88,145},40] (* Harvey P. Dale, Mar 23 2020 *)