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URL: https://oeis.org/A039758

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A039758
Triangle of B-analogs of Stirling numbers of first kind.
5
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: A123160 A378561 A109692 * A157894 A172106 A128813
KEYWORD
tabl,sign
AUTHOR
Ruedi Suter (suter(AT)math.ethz.ch)
EXTENSIONS
More terms from Petros Hadjicostas, Jul 12 2020
STATUS
approved