A267168 Growth series for affine Coxeter group B_5.

1, 6, 20, 51, 110, 211, 372, 615, 966, 1455, 2117, 2991, 4120, 5551, 7334, 9524, 12180, 15365, 19146, 23594, 28784, 34795, 41711, 49619, 58611, 68783, 80234, 93067, 107389, 123312, 140952, 160430, 181870, 205400, 231152, 259261, 289867, 323114, 359151, 398131, 440211, 485551, 534315, 586672, 642794, 702858, 767045, 835540, 908532, 986214

Offset

0,2

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

Ray Chandler, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,1,-3,4,-4,3,-1,0,0,0,-1,3,-3,1).

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=5) the G.f. is -(1+t)*(1+t+t^2+t^3)*(t^3+1)*(1+t+t^2+t^3+t^4+t^5+t^6+t^7)*(t^5+1)/(-1+t^9)/(-1+t^7)/(-1+t)^3.

a(n) = floor((32*n^4+800*n^2+131)/189 - ((2*n^2+1) mod 3)*2/27 + (((n+5) mod 7) - ((n+1) mod 7) + ((4*n^4+2*n^2+1) mod 7))/7 + (((n+8) mod 9) + ((n+6) mod 9) - ((n+2) mod 9) - (n mod 9))/9) - [n=0]. - Hoang Xuan Thanh, Jun 08 2026

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.

Sequence in context: A384743 A052515 A067117 * A266760 A213586 A119365

Adjacent sequences: A267165 A267166 A267167 * A267169 A267170 A267171

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 11 2016

Status

approved