OFFSET
0,4
COMMENTS
Triangle of B-analogs of Stirling numbers of first kind.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Priyavrat Deshpande, Krishna Menon, and Anurag Singh, Counting regions of the boxed threshold arrangement, arXiv:2101.12060 [math.CO], 2021.
Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, arXiv preprint arXiv:1510.04073 [math.PR], 2015-2017.
FORMULA
Triangle T(n, k), read by rows, given by [ -1, -2, -3, -4, -5, -6, -7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 20 2005
a(n,m) = a(n-1,m-1) - (2*n-1)*a(n-1,m) with a(n,0) = (-1)^n*(2*n-1)!! and a(n,n) = 1. - Johannes W. Meijer, Jun 08 2009
Exponential Riordan array [1/sqrt(1 + 2*x), 1/2*log(1 + 2*x)] with e.g.f. (1 + 2*x)^((t - 1)/2) = 1 + (t-1)*x + (t-1)*(t-3)*x^2/2! + .... - Peter Bala, Jun 23 2014
EXAMPLE
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9
0: 1
1: -1 1
2: 3 -4 1
3: -15 23 -9 1
4: 105 -176 86 -16 1
5: -945 1689 -950 230 -25 1
6: 10395 -19524 12139 -3480 505 -36 1
7: -135135 264207 -177331 57379 -10045 973 -49 1
8: 2027025 -4098240 2924172 -1038016 208054 -24640 1708 -64 1
9: -34459425 71697105 -53809164 20570444 -4574934 626934 -53676 2796 -81 1
...
row n = 10 :654729075 -1396704420 1094071221 -444647600 107494190 -16486680 1646778 -106800 4335 -100 1
... reformatted and extended. - Wolfdieter Lang, May 09 2017
MAPLE
nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, 0) := (-1)^n*doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1, m-1)-(2*n-1)*a(n-1, m) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 29 2012
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, 0, n}] // Flatten (* Jean-François Alcover, Oct 16 2012, after Johannes W. Meijer *)
PROG
(PARI) row(n)=Vecrev(prod(i=1, n, 'x-2*i+1)) \\ Charles R Greathouse IV, Feb 09 2017
CROSSREFS
KEYWORD
AUTHOR
Ruedi Suter (suter(AT)math.ethz.ch)
STATUS
approved