A039763 Triangle of D-analogs of Stirling numbers of first kind, rows reversed.

1, 1, 0, 1, -2, 1, 1, -6, 11, -6, 1, -12, 50, -84, 45, 1, -20, 150, -520, 809, -420, 1, -30, 355, -2100, 6439, -9390, 4725, 1, -42, 721, -6510, 33019, -92358, 127539, -62370, 1, -56, 1316, -16856, 127694, -578984, 1505524, -1984584, 945945, 1, -72, 2220, -38304, 405174, -2702448, 11228300, -27491616, 34812945, -16216200

Offset: 0

Author

Ruedi Suter (suter(AT)math.ethz.ch)

Keywords

• tabl
• sign

Comments

|T(n,k)|, 0 <= k <= n, is the number of elements in the Coxeter group D_n with absolute length k. - Jose Bastidas, Jul 16 2023

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,   0;
  1,  -2,   1;
  1,  -6,  11,   -6;
  1, -12,  50,  -84,  45;
  1, -20, 150, -520, 809, -420;
  ...
		

Links

• Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Crossrefs

Cf. A039762 (transposed triangle).

Programs

PARI

row(n) = if(n==0, [1], Vec(prod(i=1, n-1, x-2*i+1)*(x-n+1))); \\ Petros Hadjicostas, Jul 12 2020

Formula

From Petros Hadjicostas, Jul 11 2020: (Start)

T(n,k) = A039762(n,n-k) for k = 0..n.

T(n,0) = 1 for n >= 0.

T(n,n) = (-1)^n*(n-1)*(2*n-3)!! for n >= 2.

T(n,k) = [x^(n-k)] (x - (n - 1)) * Product_{k=1..n-1} (x - (2*k - 1)) for n >= 1 and k = 0..n. (End)

Extensions

More terms from Petros Hadjicostas, Jul 12 2020