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

0,4

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

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: A321627 A350557 A028338 * A136228 A356145 A154829

Adjacent sequences: A039754 A039755 A039756 * A039758 A039759 A039760

Keyword

tabl,sign,easy,nice

Author

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

Status

approved