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

%I A039758 #38 Jun 01 2025 18:05:29
%S A039758 1,1,-1,1,-4,3,1,-9,23,-15,1,-16,86,-176,105,1,-25,230,-950,1689,-945,
%T A039758 1,-36,505,-3480,12139,-19524,10395,1,-49,973,-10045,57379,-177331,
%U A039758 264207,-135135,1,-64,1708,-24640,208054,-1038016,2924172,-4098240,2027025,1,-81,2796,-53676,626934,-4574934,20570444,-53809164,71697105,-34459425
%N A039758 Triangle of B-analogs of Stirling numbers of first kind.
%C A039758 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
%H A039758 Michael De Vlieger, <a href="/A039758/b039758.txt">Table of n, a(n) for n = 0..11475</a> (rows n = 0..150, flattened)
%H A039758 Alnour Altoum, Hasan Arslan, and Mariam Zaarour, <a href="https://arxiv.org/abs/2312.14652">Cauchy numbers in type B</a>, arXiv:2312.14652 [math.CO], 2023.
%H A039758 Ruedi Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
%F A039758 T(n,k) = A039757(n,n-k). - _Petros Hadjicostas_, Jul 12 2020
%e A039758 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A039758   1;
%e A039758   1,  -1;
%e A039758   1,  -4,   3;
%e A039758   1,  -9,  23,   -15;
%e A039758   1, -16,  86,  -176,   105;
%e A039758   1, -25, 230,  -950,  1689,   -945;
%e A039758   1, -36, 505, -3480, 12139, -19524, 10395;
%e A039758 ... [Edited by _Petros Hadjicostas_, Jul 12 2020]
%t A039758 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 *)
%o A039758 (PARI) row(n)=Vec(prod(i=1, n, 'x-2*i+1)) \\ _Petros Hadjicostas_, Jul 12 2020
%Y A039758 Cf. A039757.
%K A039758 tabl,sign
%O A039758 0,5
%A A039758 Ruedi Suter (suter(AT)math.ethz.ch)
%E A039758 More terms from _Petros Hadjicostas_, Jul 12 2020