A039757 Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
1, -1, 1, 3, -4, 1, -15, 23, -9, 1, 105, -176, 86, -16, 1, -945, 1689, -950, 230, -25, 1, 10395, -19524, 12139, -3480, 505, -36, 1, -135135, 264207, -177331, 57379, -10045, 973, -49, 1, 2027025, -4098240, 2924172, -1038016, 208054, -24640, 1708, -64, 1, -34459425, 71697105, -53809164, 20570444, -4574934, 626934, -53676, 2796, -81, 1
Offset: 0
Examples
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
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.
Crossrefs
Programs
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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
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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 *) -
PARI
row(n)=Vecrev(prod(i=1,n,'x-2*i+1)) \\ Charles R Greathouse IV, Feb 09 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
Comments