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Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
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%I #52 May 07 2021 19:53:41

%S 1,-1,1,3,-4,1,-15,23,-9,1,105,-176,86,-16,1,-945,1689,-950,230,-25,1,

%T 10395,-19524,12139,-3480,505,-36,1,-135135,264207,-177331,57379,

%U -10045,973,-49,1,2027025,-4098240,2924172,-1038016,208054,-24640,1708,-64,1,-34459425,71697105,-53809164,20570444,-4574934,626934,-53676,2796,-81,1

%N Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).

%C Triangle of B-analogs of Stirling numbers of first kind.

%H Michael De Vlieger, <a href="/A039757/b039757.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)

%H Priyavrat Deshpande, Krishna Menon, and Anurag Singh, <a href="https://arxiv.org/abs/2101.12060">Counting regions of the boxed threshold arrangement</a>, arXiv:2101.12060 [math.CO], 2021.

%H 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.

%H Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, <a href="http://arxiv.org/abs/1510.04073">Convex hulls of random walks, hyperplane arrangements, and Weyl chambers</a>, arXiv preprint arXiv:1510.04073 [math.PR], 2015-2017.

%F 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

%F 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

%F 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

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9

%e 0: 1

%e 1: -1 1

%e 2: 3 -4 1

%e 3: -15 23 -9 1

%e 4: 105 -176 86 -16 1

%e 5: -945 1689 -950 230 -25 1

%e 6: 10395 -19524 12139 -3480 505 -36 1

%e 7: -135135 264207 -177331 57379 -10045 973 -49 1

%e 8: 2027025 -4098240 2924172 -1038016 208054 -24640 1708 -64 1

%e 9: -34459425 71697105 -53809164 20570444 -4574934 626934 -53676 2796 -81 1

%e ...

%e row n = 10 :654729075 -1396704420 1094071221 -444647600 107494190 -16486680 1646778 -106800 4335 -100 1

%e ... reformatted and extended. - _Wolfdieter Lang_, May 09 2017

%p 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

%t 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_ *)

%o (PARI) row(n)=Vecrev(prod(i=1,n,'x-2*i+1)) \\ _Charles R Greathouse IV_, Feb 09 2017

%Y A028338 is unsigned version.

%Y From _Johannes W. Meijer_, Jun 08 2009: (Start)

%Y A161198 is an unsigned scaled triangle version.

%Y A109692 is an unsigned transposed triangle version.

%Y A000007 equals the row sums. (End)

%Y A000165(n)*(-1)^n (alternating row sums).

%K tabl,sign,easy,nice

%O 0,4

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