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 A039757 Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)). 12
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Triangle of B-analogs of Stirling numbers of first kind. LINKS 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 A028338 is unsigned version. From Johannes W. Meijer, Jun 08 2009: (Start) A161198 is an unsigned scaled triangle version. A109692 is an unsigned transposed triangle version. A000007 equals the row sums. (End) A000165(n)*(-1)^n (alternating row sums). Sequence in context: A303728 A321627 A028338 * A136228 A154829 A215241 Adjacent sequences:  A039754 A039755 A039756 * A039758 A039759 A039760 KEYWORD tabl,sign,easy,nice AUTHOR Ruedi Suter (suter(AT)math.ethz.ch) STATUS approved

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Last modified April 19 09:52 EDT 2021. Contains 343110 sequences. (Running on oeis4.)