cp's OEIS Frontend

A039757 Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).

%I A039757 #52 May 07 2021 19:53:41
%S A039757 1,-1,1,3,-4,1,-15,23,-9,1,105,-176,86,-16,1,-945,1689,-950,230,-25,1,
%T A039757 10395,-19524,12139,-3480,505,-36,1,-135135,264207,-177331,57379,
%U A039757 -10045,973,-49,1,2027025,-4098240,2924172,-1038016,208054,-24640,1708,-64,1,-34459425,71697105,-53809164,20570444,-4574934,626934,-53676,2796,-81,1
%N A039757 Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
%C A039757 Triangle of B-analogs of Stirling numbers of first kind.
%H A039757 Michael De Vlieger, <a href="/A039757/b039757.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A039757 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 A039757 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 A039757 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 A039757 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 A039757 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 A039757 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 A039757 The triangle T(n, k) begins:
%e A039757 n\k        0        1         2        3        4      5      6    7   8  9
%e A039757 0:         1
%e A039757 1:        -1        1
%e A039757 2:         3       -4         1
%e A039757 3:       -15       23        -9        1
%e A039757 4:       105     -176        86      -16        1
%e A039757 5:      -945     1689      -950      230      -25      1
%e A039757 6:     10395   -19524     12139    -3480      505    -36      1
%e A039757 7:   -135135   264207   -177331    57379   -10045    973    -49    1
%e A039757 8:   2027025 -4098240   2924172 -1038016   208054 -24640   1708  -64   1
%e A039757 9: -34459425 71697105 -53809164 20570444 -4574934 626934 -53676 2796 -81  1
%e A039757 ...
%e A039757 row n = 10 :654729075 -1396704420 1094071221 -444647600 107494190 -16486680 1646778 -106800 4335 -100 1
%e A039757 ... reformatted and extended. - _Wolfdieter Lang_, May 09 2017
%p A039757 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 A039757 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 A039757 (PARI) row(n)=Vecrev(prod(i=1,n,'x-2*i+1)) \\ _Charles R Greathouse IV_, Feb 09 2017
%Y A039757 A028338 is unsigned version.
%Y A039757 From _Johannes W. Meijer_, Jun 08 2009: (Start)
%Y A039757 A161198 is an unsigned scaled triangle version.
%Y A039757 A109692 is an unsigned transposed triangle version.
%Y A039757 A000007 equals the row sums. (End)
%Y A039757 A000165(n)*(-1)^n (alternating row sums).
%K A039757 tabl,sign,easy,nice
%O A039757 0,4
%A A039757 Ruedi Suter (suter(AT)math.ethz.ch)