A039758 Triangle of B-analogs of Stirling numbers of first kind.

1, 1, -1, 1, -4, 3, 1, -9, 23, -15, 1, -16, 86, -176, 105, 1, -25, 230, -950, 1689, -945, 1, -36, 505, -3480, 12139, -19524, 10395, 1, -49, 973, -10045, 57379, -177331, 264207, -135135, 1, -64, 1708, -24640, 208054, -1038016, 2924172, -4098240, 2027025, 1, -81, 2796, -53676, 626934, -4574934, 20570444, -53809164, 71697105, -34459425

Offset

0,5

Comments

Triangle T(n,k), read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [ -1, -2, -3, -4, -5, -6, -7, -8, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 08 2005

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)

Alnour Altoum, Hasan Arslan, and Mariam Zaarour, Cauchy numbers in type B, arXiv:2312.14652 [math.CO], 2023.

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

Formula

T(n,k) = A039757(n,n-k). - Petros Hadjicostas, Jul 12 2020

Example

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  -1;
  1,  -4,   3;
  1,  -9,  23,   -15;
  1, -16,  86,  -176,   105;
  1, -25, 230,  -950,  1689,   -945;
  1, -36, 505, -3480, 12139, -19524, 10395;
... [Edited by Petros Hadjicostas, Jul 12 2020]

Mathematica

a[n_, m_] := a[n, m] = a[n - 1, m - 1] - (2*n - 1)*a[n - 1, m]; a[n_, 0] := (-1)^n*(2*n - 1)!!; a[n_, n_] = 1; Table[a[n, m], {n, 0, 9}, {m, n, 0, -1}]] // Flatten (* Michael De Vlieger, Dec 29 2023, after Jean-François Alcover at A039757 *)

Prog

(PARI) row(n)=Vec(prod(i=1, n, 'x-2*i+1)) \\ Petros Hadjicostas, Jul 12 2020

Crossrefs

Cf. A039757.

Sequence in context: A214859 A123160 A109692 * A157894 A172106 A128813

Adjacent sequences: A039755 A039756 A039757 * A039759 A039760 A039761

Keyword

tabl,sign

Author

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

Extensions

More terms from Petros Hadjicostas, Jul 12 2020

Status

approved