A161733 -id:A161733 - cp's OEIS frontend

A008137 Coordination sequence T1 for Zeolite Code LTA and RHO.

1, 4, 9, 17, 28, 42, 60, 81, 105, 132, 162, 196, 233, 273, 316, 362, 412, 465, 521, 580, 642, 708, 777, 849, 924, 1002, 1084, 1169, 1257, 1348, 1442, 1540, 1641, 1745, 1852, 1962, 2076, 2193, 2313, 2436, 2562, 2692, 2825, 2961, 3100, 3242, 3388, 3537, 3689

Offset: 0

Ralf W. Grosse-Kunstleve

Also, growth series for the affine Coxeter (or Weyl) groups B_3. - N. J. A. Sloane, Jan 11 2016

Also, coordination sequence for "rho" 3D uniform tiling. - N. J. A. Sloane, Feb 10 2018

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).
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tilings #25 and 27.
W. M. Meier, D. H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996.

R. W. Grosse-Kunstleve, Table of n, a(n) for n = 0..1000
R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences
R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.
Sean A. Irvine, Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane
International Zeolite Association, Database of Zeolite Structures
Reticular Chemistry Structure Resource (RCSR), The lta tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The rho tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

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.
For partial sums see A299276.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

(1-x^2)*(1-x^4)*(1-x^6)/((1-x)^4*(1-x^3)*(1-x^5));
seq(coeff(series(%,x,n+1),x,n), n=0..48);

a(5*m+k) = 40*m^2 + 16*k*m + one of 5 numbers depending on k, 0 <= k < 5 (N. J. A. Sloane).
G.f.: (1-x^2)*(1-x^4)*(1-x^6)/((1-x)^4*(1-x^3)*(1-x^5)). This can also be written as (x+1)^3*(x^2+1)*(x^2-x+1)/((1-x)^3*(x^4+x^3+x^2+x+1)). - N. J. A. Sloane, Feb 10 2018
a(n) = 12/5 - 0^n + (8/5)*n^2 - (1/25)*(5+sqrt(5))*cos(2*Pi*n/5) - (1/25)*(5-sqrt(5))*cos(4*Pi*n/5). - Eric Simon Jacob, Feb 12 2023

A128084 Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 5, 7, 8, 8, 7, 5, 3, 1, 1, 4, 9, 16, 24, 32, 39, 44, 46, 44, 39, 32, 24, 16, 9, 4, 1, 1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919

Offset: 0

Paul D. Hanna, Feb 14 2007

See A128080 for the triangle of coefficients of q in the q-analog of the odd double factorials.

Row maxima ~ 2^n*n!/(sigma * sqrt(2*Pi)), sigma^2 = (4*n^3 + 6*n^2 - n)/36 = variance of Coxeter group B_n (see also A161858). - Mikhail Gaichenkov, Feb 08 2023

			The row sums form A000165, the even double factorial numbers:
  [1, 2, 8, 48, 384, 3840, 46080, 645120, ..., (2n)!!, ...].
Triangle begins:
 1;
 1, 1;
 1, 2,  2,  2,  1;
 1, 3,  5,  7,  8,  8,   7,   5,   3,   1;
 1, 4,  9, 16, 24, 32,  39,  44,  46,  44,  39,  32,  24,  16,   9,   4,   1;
 ...

Paul D. Hanna, Rows n=0..30 of triangle, in flattened form.
Hasan Arslan, A combinatorial interpretation of Mahonian numbers of type B, arXiv:2404.05099 [math.CO], 2024.
Thomas Kahle and Christian Stump, Counting inversions and descents of random elements in finite Coxeter groups, arXiv:1802.01389 [math.CO], 2018-2019.
Ali Kessouri, Moussa Ahmia, Hasan Arslan, and Salim Mesbahi, Combinatorics of q-Mahonian numbers of type B and log-concavity, arXiv:2408.02424 [math.CO], 2024. See p. 6.
Eric Weisstein's World of Mathematics, q-Factorial.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
A. V. Yurkin, New binomial and new view on light theory, LAP Lambert Academic Publishing, 2013, 78 pages.

Cf. A000165 ((2n)!!); A128085 (central terms); A128086 (diagonal), A128087 (row squared sums); A128080, A002522 (row lengths).

The growth series for the affine Coxeter groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858.

t[n_, k_] := If[k < 0 || k > n^2, 0, If[n == 0, 1, Coefficient[ Series[ Product[ (1 - q^(2*j))/(1 - q), {j, 1, n}], {q, 0, n^2}], q, k]]]; Table[t[n, k], {n, 0, 6}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Mar 06 2013, translated from Pari *)

{T(n,k) = if(k<0||k>n^2,0, if(n==0,1, polcoeff( prod(j=1,n,(1-q^(2*j))/(1-q)), k,q) ))}
for(n=0,8,for(k=0,n^2,print1(T(n,k),", "));print(""))

row(n)=Vec(prod(j=1, n, (1-x^(2*j))/(1-x))) \\ Andrew Howroyd, Mar 21 2025

A008576 Coordination sequence for planar net 4.8.8.

Original entry on oeis.org

1, 3, 5, 8, 11, 13, 16, 19, 21, 24, 27, 29, 32, 35, 37, 40, 43, 45, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 99, 101, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 131, 133

Offset: 0

N. J. A. Sloane

Also, growth series for the affine Coxeter (or Weyl) groups B_2. - N. J. A. Sloane, Jan 11 2016

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).
A. V. Shutov, On the number of words of a given length in plane crystallographic groups (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1.

T. D. Noe, Table of n, a(n) for n = 0..1000
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011
Agnes Azzolino, Illustration of 4.8.8 tiling [From previous link]
Jillian Cervantes and Pamela E. Harris, (t,r) Broadcast Domination Numbers and Densities of the Truncated Square Tiling Graph, arXiv:2408.13331 [math.CO], 2024. See p. 8.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.5 p. 22.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
W. M. Meier and H. J. Moeck, Topology of 3-D 4-connected nets ..., J. Solid State Chem 27 1979 349-355, esp. p. 351.
Reticular Chemistry Structure Resource, fes
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A042965, A102283.
List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
For partial sums see A008577.
The growth series for the finite Coxeter (or Weyl) groups B_3 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.

if n mod 3 = 0 then 8*n/3 elif n mod 3 = 1 then 8*(n-1)/3+3 else 8*(n-2)/3+5 fi;

cspn[n_]:=Module[{c=Mod[n,3]},Which[c==0,(8n)/3,c==1,(8(n-1))/3+3,True,(8(n-2))/3+5]]; Join[{1},Array[cspn,50]] (* or *) Join[{1}, LinearRecurrence[ {1,0,1,-1},{3,5,8,11},50]] (* Harvey P. Dale, Nov 24 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,1,0,1]^n*[1;3;5;8])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

G.f.: ((1+x)^2*(1+x^2))/((1-x)^2*(1+x+x^2)). - Ralf Stephan, Apr 24 2004
a(0)=1, a(1)=3, a(2)=5, a(3)=8, a(4)=11, a(n) = a(n-1) + a(n-3) - a(n-4). - Harvey P. Dale, Nov 24 2011
a(0)=1; thereafter a(3k)=8k, a(3k+1)=8k+3, a(3k+2)=8k+5. - N. J. A. Sloane, Dec 22 2015
The above g.f. and recurrence were originally empirical observations, but I now have a proof (details will be added later). This also justifies the Maple and Mma programs and the b-file. - N. J. A. Sloane, Dec 22 2015
Sum of alternate terms of A042965 (numbers not congruent to 2 mod 4), such that A042965(n) = A042965(n+1) + A042965(n-1). - Gary W. Adamson, Sep 12 2007
a(n) = (2/9)*(12*n + (3/2)*A102283(n)) for n > 0. - Stefano Spezia, Aug 07 2022

A161696 Number of reduced words of length n in the Weyl group B_3.

Original entry on oeis.org

1, 3, 5, 7, 8, 8, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

nonn

If the zeros are ignored, this is the coordination sequence for the truncated cuboctahedron (see the Karzes link). - N. J. A. Sloane, Jan 08 2020

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Tom Karzes, Polyhedron Coordination Sequences

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.

m:=10; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..3]])/(1-t)^3)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..3),x,n+1), x, n), n = 0 .. 120); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,3}] /(1-x)^3, {x,0,9}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^10); Vec(prod(k=1,3,1-t^(2*k))/(1-t)^3) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161699 Number of reduced words of length n in the Weyl group B_6.

Original entry on oeis.org

1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919, 2254, 2565, 2832, 3037, 3166, 3210, 3166, 3037, 2832, 2565, 2254, 1919, 1580, 1255, 958, 699, 484, 315, 190, 104, 50, 20, 6, 1

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

G. C. Greubel, Table of n, a(n) for n = 0..36

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.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..6]])/(1-t)^6)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..6),x,n+1), x, n), n = 0 .. 36); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) / (1 - x)^6, {x, 0, 50}], x]  (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,6,1-t^(2*k))/(1-t)^6) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Product_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161716 Number of reduced words of length n in the Weyl group B_7.

Original entry on oeis.org

1, 7, 27, 77, 181, 371, 686, 1170, 1869, 2827, 4082, 5662, 7581, 9835, 12399, 15225, 18242, 21358, 24464, 27440, 30162, 32510, 34376, 35672, 36336, 36336, 35672, 34376, 32510, 30162, 27440, 24464, 21358, 18242, 15225, 12399, 9835, 7581, 5662

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

G. C. Greubel, Table of n, a(n) for n = 0..49

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.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..7]])/(1-t)^7)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..7),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) / (1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^50); Vec(prod(k=1,7,1-t^(2*k))/(1-t)^7) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Product_{k=1..m} (1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161717 Number of reduced words of length n in the Weyl group B_8.

Original entry on oeis.org

1, 8, 35, 112, 293, 664, 1350, 2520, 4389, 7216, 11298, 16960, 24541, 34376, 46775, 62000, 80241, 101592, 126029, 153392, 183373, 215512, 249202, 283704, 318171, 351680, 383270, 411984, 436913, 457240, 472281, 481520, 484636, 481520, 472281

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Vincenzo Librandi, Table of n, a(n) for n = 0..64

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.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..8]])/(1-t)^8)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..8),x,65), x, n), n = 0 .. 64); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) / (1 - x)^8, {x, 0, 70}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,8,1-t^(2*k))/(1-t)^8) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Product_{k=1..m} (1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161755 Number of reduced words of length n in the Weyl group B_10.

Original entry on oeis.org

1, 10, 54, 210, 659, 1772, 4235, 9218, 18590, 35178, 63064, 107910, 177297, 281060, 431598, 644136, 936915, 1331286, 1851685, 2525468, 3382588, 4455100, 5776486, 7380800, 9301642, 11570980, 14217849, 17266966, 20737309, 24640716, 28980565

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

G. C. Greubel, Table of n, a(n) for n = 0..100

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.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..10]])/(1-t)^10)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..10),x,101), x, n), n = 0 .. 100); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[(1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) (1 - x^18) (1 - x^20) / (1 - x)^10, {x, 0, 100}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,10,1-t^(2*k))/(1-t)^10) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Product_{k=1..m} (1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161776 Number of reduced words of length n in the Weyl group B_11.

Original entry on oeis.org

1, 11, 65, 275, 934, 2706, 6941, 16159, 34749, 69927, 132991, 240901, 418198, 699258, 1130856, 1774992, 2711907, 4043193, 5894878, 8420346, 11802934, 16258034, 22034519, 29415309, 38716897, 50287667, 64504857, 81770051

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

G. C. Greubel, Table of n, a(n) for n = 0..121

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.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..11]])/(1-t)^11)); // G. C. Greubel, Oct 24 2018

seq(coeff(series(mul((1-x^(2*k))/(1-x),k=1..11),x,122), x, n), n = 0 .. 121); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[((1 - x^2) (1 - x^4) (1 - x^6) (1 - x^8) (1 - x^10) (1 - x^12) (1 - x^14) (1 - x^16) (1 - x^18) (1 - x^20) (1 - x^22)) / (1 - x)^11, {x, 0, 121}], x] (* Vincenzo Librandi, Aug 22 2016 *)

t='t+O('t^40); Vec(prod(k=1,11,1-t^(2*k))/(1-t)^11) \\ G. C. Greubel, Oct 24 2018

G.f. for B_m is the polynomial Prod_{k=1..m} (1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A161858 Number of reduced words of length n in the Weyl group B_12.

Original entry on oeis.org

1, 12, 77, 352, 1286, 3992, 10933, 27092, 61841, 131768, 264759, 505660, 923858, 1623116, 2753972, 4528964, 7240871, 11284064, 17178942, 25599288, 37402222, 53660256, 75694775, 105110084, 143826980, 194114636, 258619428, 340389204, 442891395, 570023312, 726112969

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

G. C. Greubel, Table of n, a(n) for n = 0..144

Row n=12 of A128084.

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..12]])/(1-t)^12)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..12),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)),{k,1,12}]/(1-x)^12,{x,0,50}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^50); Vec(prod(k=1,12,1-t^(2*k))/(1-t)^12) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Product_{k=1..m} (1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

cp's OEIS Frontend

A008137 Coordination sequence T1 for Zeolite Code LTA and RHO.

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A128084 Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

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A008576 Coordination sequence for planar net 4.8.8.

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A161696 Number of reduced words of length n in the Weyl group B_3.

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A161699 Number of reduced words of length n in the Weyl group B_6.

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A161716 Number of reduced words of length n in the Weyl group B_7.

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A161717 Number of reduced words of length n in the Weyl group B_8.