A067318 -id:A067318 - cp's OEIS frontend

A197131 Sum of the reflection (absolute) lengths of all elements in the Coxeter group of type D_n.

4, 46, 544, 7216, 108096, 1816704, 33951744, 699512832, 15765626880, 386046443520, 10208951009280, 290039357767680, 8811722692362240, 285113464809062400, 9789232245217689600, 355501479519741542400, 13615286053738276454400, 548476851979845579571200

Offset: 2

Cathy Kriloff, Oct 10 2011

			a(3)=46 because W(D_3)=W(A_3) and in sequence A067318, a(3)=46.

P. Renteln, The distance spectra of Cayley graphs of Coxeter groups, Discrete Math., 311 (2011), 738-755.

B. Foster-Greenwood, C. Kriloff, Spectra of Cayley Graphs of Complex Reflection Groups, arXiv preprint arXiv:1502.07392, 2015

Cf. A067318, A197130

seq(2^(n-1)*factorial(n)*(add((2*k-1)/(2*k), k=1..n-1)+(n-1)/n), n=2..100);

Table[2^(n-1)*Factorial[n]*(Sum[(2*k-1)/(2*k),{k,1,n-1}]+(n-1)/n), {n,2,100}]

[2^(n-1)*factorial(n)*(sum([(2*k-1)/(2*k) for k in [1..n-1]])+(n-1)/n) for n in [2..100]]

a(n)=Sum_{w in W(D_n)} l_T(w)=|W(D_n)|Sum_{i=1}^n (d_i-1)/d_i=2^(n-1)*n!*(1/2+3/4+...+(2n-3)/(2n-2)+(n-1)/n) where T=all reflections in W(D_n), l_T(1)=0 and otherwise l_T(w)=min{k|w=t_1*...*t_k for t_i in T}, and d_1,...,d_n are the degrees of W(D_n)

A371762 Triangle read by rows: the polynomial coefficients of the numerator of the rational solution of the linear recurrence equations of the rows of A371761.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 7, 6, 0, 1, 19, 46, 24, 0, 1, 46, 251, 326, 120, 0, 1, 104, 1163, 3016, 2556, 720, 0, 1, 225, 4831, 23283, 35848, 22212, 5040, 0, 1, 473, 18523, 158531, 417148, 437228, 212976, 40320, 0, 1, 976, 66886, 976636, 4285549, 7084804, 5586444, 2239344, 362880

Offset: 0

Peter Luschny, Apr 06 2024

Let R(n) = N(n)/D(n) denote the ordinary rational generating function of row n of A371761 as given by its linear recurrence equation. N(n) is the row polynomial Sum_{k=0..n} T(n, k)*x^k and D(n) = Sum_{k=0..n} Stirling1(n+1, n+1-k)*x^k. Thus A371761(n, k) = [x^k] N(n)/D(n).

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,   2;
  [3] 0, 1,   7,     6;
  [4] 0, 1,  19,    46,     24;
  [5] 0, 1,  46,   251,    326,     120;
  [6] 0, 1, 104,  1163,   3016,    2556,     720;
  [7] 0, 1, 225,  4831,  23283,   35848,   22212,    5040;
  [8] 0, 1, 473, 18523, 158531,  417148,  437228,  212976,   40320;
  [9] 0, 1, 976, 66886, 976636, 4285549, 7084804, 5586444, 2239344, 362880;
.
The rational generating function for row 3 of A371761 is:
gf = (6*x^3 + 7*x^2 + x)/(-6*x^3 + 11*x^2 - 6*x + 1).

Cf. A029767 (row sums), A000142 (main diagonal), A067318 (subdiagonal).

Cf. A371761.

cp's OEIS Frontend

A197131 Sum of the reflection (absolute) lengths of all elements in the Coxeter group of type D_n.

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