A128084 -id:A128084 - cp's OEIS frontend

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

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.

A267167 Growth series for affine Coxeter group B_4.

Original entry on oeis.org

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

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).

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).

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)*(1+t^3)*(1-t^4)*(1-t^8)/((1-t)^5*(1-t^5)*(1 - t^7)))); // G. C. Greubel, Oct 24 2018

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

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 *)

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

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

A267175 Growth series for affine Coxeter group B_12.

Original entry on oeis.org

1, 13, 90, 443, 1741, 5811, 17109, 45577, 111852, 256282, 553866, 1138110, 2237924, 4233126, 7735923, 13707967, 23625303, 39706809, 65225654, 104927954, 165588279, 256738054, 391610309, 588352779, 871571154, 1274275456, 1840315206, 2627403376, 3710845242, 5188106314, 7184373674, 9859287465, 13415044111, 18106100284, 24250736849

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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).

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 82, -108, 53, 90, -225, 217, -27, -217, 306, -144, -133, 261, -99, -217, 379, -197, -188, 404, -207, -252, 542, -360, -154, 521, -387, -107, 459, -326, -117, 358, -88, -452, 693, -327, -342, 666, -296, -434, 800, -415, -349, 720, -315, -460, 811, -359, -457, 800, -277, -649, 1102, -649, -277, 800, -457, -359, 811, -460, -315, 720, -349, -415, 800, -434, -296, 666, -342, -327, 693, -452, -88, 358, -117, -326, 459, -107, -387, 521, -154, -360, 542, -252, -207, 404, -188, -197, 379, -217, -99, 261, -133, -144, 306, -217, -27, 217, -225, 90, 53, -108, 82, -36, 9, -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.

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].

A128080 Triangle, read by rows of n(n-1)+1 terms, of coefficients of q in the q-analog of the odd double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 6, 9, 12, 14, 15, 14, 12, 9, 6, 3, 1, 1, 4, 10, 19, 31, 45, 60, 74, 86, 94, 97, 94, 86, 74, 60, 45, 31, 19, 10, 4, 1, 1, 5, 15, 34, 65, 110, 170, 244, 330, 424, 521, 614, 696, 760, 801, 815, 801, 760, 696, 614, 521, 424, 330, 244, 170

Offset: 0

Paul D. Hanna, Feb 14 2007

See A128084 for the triangle of coefficients of q in the q-analog of the even double factorials.

			Triangle begins:
  1;
  1;
  1,1,1;
  1,2,3,3,3,2,1;
  1,3,6,9,12,14,15,14,12,9,6,3,1;
  1,4,10,19,31,45,60,74,86,94,97,94,86,74,60,45,31,19,10,4,1;
  ...

Paul D. Hanna, Rows n=0..31 of triangle, in flattened form.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A001147 (row sums); A128081 (central terms), A128082 (diagonal), A128083 (row squared sums); A128084.

b:= proc(n) option remember; `if`(n=0, 1,
      simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
    end:
T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 22 2021

Catenate@Table[CoefficientList[Cancel@FunctionExpand[-q QPochhammer[1/q, q^2, n + 1]/(1 - q)^(n + 1)], q], {n, 0, 6}] (* Vladimir Reshetnikov, Sep 22 2021 *)
T[n_] := If[n == 0, {1}, Product[(1 - q^(2 j - 1))/(1 - q), {j, 1, n}] + O[q]^(n (n + 1)) // CoefficientList[#, q]&];
Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Sep 27 2022 *)

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

The row sums are A001147, the odd double factorial numbers (2n-1)!!.

A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).

Original entry on oeis.org

1, 1, 2, 8, 46, 340, 3210, 36336, 484636, 7394458, 127707302, 2454109404, 52091631896, 1207854671388, 30431260261770, 826657521349952, 24114046688034516, 751085176539860458, 24899882719111953556

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

See A128081 for central coefficients of q in the q-analog of the odd double factorials. Also, A000140 is the central coefficients of q-factorials, giving the maximum number of permutations on n letters having the same number of inversions.

			a(n) is the central term of the q-analog of even double factorials,
in which the coefficients of q (triangle A128084) begin:
n=0: (1);
n=1: (1),1;
n=2: 1,2,(2),2,1;
n=3: 1,3,5,7,(8),8,7,5,3,1;
n=4: 1,4,9,16,24,32,39,44,(46),44,39,32,24,16,9,4,1;
n=5: 1,5,14,30,54,86,125,169,215,259,297,325,(340),340,325,297,...;...
The terms enclosed in parenthesis are initial terms of this sequence.

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A000165 ((2n)!!); A128084 (triangle), A128086 (diagonal); A128081.

a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-q^(2*k))/(1-q)),n^2\2,q))

A128596 Triangle, read by rows, of coefficients of q^(nk) in the q-analog of the even double factorials: T(n,k) = [q^(nk)] 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, 1, 1, 7, 7, 1, 1, 24, 46, 24, 1, 1, 86, 297, 297, 86, 1, 1, 315, 1919, 3210, 1919, 315, 1, 1, 1170, 12399, 32510, 32510, 12399, 1170, 1, 1, 4389, 80241, 318171, 484636, 318171, 80241, 4389, 1, 1, 16588, 520399, 3054100, 6730832, 6730832

Offset: 0

Paul D. Hanna, Mar 12 2007

			Row sums equal 2*A000165(n-1) for n>0, twice the even double factorials:
[1, 2, 4, 16, 96, 768, 7680, 92160, 1290240, ..., 2*(2n-2)!!, ...].
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 7, 7, 1;
1, 24, 46, 24, 1;
1, 86, 297, 297, 86, 1;
1, 315, 1919, 3210, 1919, 315, 1;
1, 1170, 12399, 32510, 32510, 12399, 1170, 1;
1, 4389, 80241, 318171, 484636, 318171, 80241, 4389, 1;
1, 16588, 520399, 3054100, 6730832, 6730832, 3054100, 520399, 16588, 1;
1, 63064, 3382588, 28980565, 89514691, 127707302, 89514691, 28980565, 3382588, 63064, 1;

Cf. A128084; A000165 ((2n)!!); A128086 (column 1), A128597 (column 2), A128598 (column 3); variant: A128592.

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)),n*k,q)))

T(n,k) = A128084(n,nk) where A128084 is the triangle of coefficients of q in the q-analog of the even double factorials.

A128087 Sum of squared coefficients of q in the q-analog of the even double factorials.

Original entry on oeis.org

1, 2, 14, 296, 12938, 956720, 107245250, 16966970200, 3601980861720, 988252809411908, 340375635448973106, 143798619953044471444, 73123320014581106403732, 44060303354020797873285800

Offset: 0

Paul D. Hanna, Feb 14 2007

nonn

See A128083 for sum of squared coefficients of q in the q-analog of the odd double factorials. Also, A127728 is the sum of squared coefficients of q in the q-factorials.

Cf. A000165 ((2n)!!); A128084 (triangle), A128085 (central terms), A128086 (diagonal); A127728.

{a(n)=if(n==0,1,sum(k=0,n^2,polcoeff(prod(j=1,n,(1-q^(2*j))/(1-q)),k,q)^2))}

A128597 Column 2 of triangle A128596; a(n) = coefficient of q^(2n+4) in the q-analog of the even double factorials (2n+4)!! for n>=0.

Original entry on oeis.org

1, 7, 46, 297, 1919, 12399, 80241, 520399, 3382588, 22034519, 143826980, 940569228, 6161492611, 40426009162, 265617089899, 1747501590554, 11510584144337, 75901841055650, 501007227527884, 3310076954166501

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

Cf. A128596; A128084; A000165 ((2n)!!); A128086 (column 1), A128598 (column 3).

{a(n)=polcoeff(prod(j=1,n+2,(1-q^(2*j))/(1-q)),2*n+4,q)}

a(n) = [q^(2n+4)] Product_{j=1..n+2} (1-q^(2j))/(1-q) for n>=0.

A128598 Column 3 of triangle A128596; a(n) = coefficient of q^(3n+9) in the q-analog of the even double factorials (2n+6)!! for n>=0.

Original entry on oeis.org

1, 24, 297, 3210, 32510, 318171, 3054100, 28980565, 273077443, 2562036673, 23973009386, 223949654108, 2090070431683, 19496003736658, 181815760387221, 1695523268254637, 15813185728272754, 147508341317700463

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

Cf. A128596; A128084; A000165 ((2n)!!); A128086 (column 1), A128597 (column 2).

{a(n)=polcoeff(prod(j=1,n+3,(1-q^(2*j))/(1-q)),3*n+9,q)}

a(n) = [q^(3n+9)] Product_{j=1..n+3} (1-q^(2j))/(1-q) for n>=0.

A161879 Number of reduced words of length n in the Weyl group B_19.

Original entry on oeis.org

1, 19, 189, 1311, 7124, 32300, 127091, 445721, 1420364, 4172476, 11426240, 29429784, 71808030, 166970290, 371826581, 796341623, 1646167391, 3294638285, 6401307860, 12102626404, 22312161586, 40184022430, 70815181390, 122291804610, 207223417349, 344959019207, 564743768579

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 alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Andrew Howroyd, Table of n, a(n) for n = 0..361

Row n=19 of A128084.

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.

cp's OEIS Frontend

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

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A267167 Growth series for affine Coxeter group B_4.

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A267175 Growth series for affine Coxeter group B_12.

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A128080 Triangle, read by rows of n(n-1)+1 terms, of coefficients of q in the q-analog of the odd double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

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A128085 Central coefficients of q in the q-analog of the even double factorials: a(n) = [q^([n^2/2])] Product_{j=1..n} (1-q^(2j))/(1-q).

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A128596 Triangle, read by rows, of coefficients of q^(nk) in the q-analog of the even double factorials: T(n,k) = [q^(nk)] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

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A128087 Sum of squared coefficients of q in the q-analog of the even double factorials.

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A128597 Column 2 of triangle A128596; a(n) = coefficient of q^(2n+4) in the q-analog of the even double factorials (2n+4)!! for n>=0.

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