A267167 Growth series for affine Coxeter group B_4.

1, 5, 14, 31, 59, 101, 161, 243, 351, 488, 658, 865, 1112, 1403, 1741, 2130, 2574, 3077, 3643, 4274, 4974, 5747, 6597, 7528, 8543, 9646, 10840, 12129, 13517, 15007, 16603, 18309, 20129, 22066, 24123, 26304, 28613, 31054, 33631, 36347, 39205, 42209, 45363, 48671, 52136, 55762, 59553, 63512, 67643, 71949, 76434, 81102, 85957, 91003, 96242

Offset: 0

Author

N. J. A. Sloane, Jan 11 2016

Keywords

• nonn
• easy

References

• N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,2,-2,1,0,0,-1,2,-1).

Crossrefs

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

Programs

Magma

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-t^2)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7)))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series((1-x^2)*(1+x^3)*(1-x^4)*(1-x^8)/((1-x)^5*(1-x^5)*(1-x^7)),x,n+1), x, n), n = 0 .. 55); # Muniru A Asiru, Oct 25 2018

Mathematica

CoefficientList[Series[(1-t^2)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7)), {t, 0, 50}], t] (* G. C. Greubel, Oct 24 2018 *)

PARI

t='t+O('t^40); Vec((1-t^2)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7))) \\ G. C. Greubel, Oct 24 2018

Formula

The growth series for the affine Coxeter group of type B_k (k >= 2) has g.f. = Product_i (1-x^{m_i+1})/((1-x)*(1-x^{m_i})) where the m_i are [1,3,5,...,2k-1].

Here (k=4) the G.f. is (1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(t^3+1)*(1+t+t^2+t^3)*(1+t) / (-1+t^7)/(-1+t^5)/(-1+t)^2.

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9) - a(n-12) + 2*a(n-13) - a(n-14), n > 0. - Muniru A Asiru, Oct 25 2018