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

A028338 → x ↦ −x (shift -1)

A039757 only rational roots

Proved for all n: all roots real, in [0, ∞) · alternating coefficient signs · interlacing (weak) what this means

Recurrence from OEIS A039757

\(P_{n}(x) = \left(3 + x - 2\,n\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-1	1							0
3	3	-4	1						0
4	-15	23	-9	1					0
5	105	-176	86	-16	1				0
6	-945	1689	-950	230	-25	1			0
7	10395	-19524	12139	-3480	505	-36	1		0
8	-135135	264207	-177331	57379	-10045	973	-49	1	0

Show all 15 stored rows

Generated b-files: A039757, A039758 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-17f1d12052bd159d · revision 1 · JSON
Definition source: oeis_catalog:A039757
Fingerprint: f6-cb277eccfb2eac40

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A039757: triangle, 55 terms agree, 10 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A039758: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:        -1,        1;
   3:         3,       -4,         1;
   4:       -15,       23,        -9,        1;
   5:       105,     -176,        86,      -16,        1;
   6:      -945,     1689,      -950,      230,      -25,      1;
   7:     10395,   -19524,     12139,    -3480,      505,    -36,      1;
   8:   -135135,   264207,   -177331,    57379,   -10045,    973,    -49,    1;
   9:   2027025, -4098240,   2924172, -1038016,   208054, -24640,   1708,  -64,   1;
  10: -34459425, 71697105, -53809164, 20570444, -4574934, 626934, -53676, 2796, -81, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(2*n-3-x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2*n - 3 - x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(2*n - 3 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A039757: triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  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;

%F Formula

The row polynomials satisfy P(n,x) = -(2*n-1-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2*n - 1 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(2*n - 1 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A039758: reversed triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  -1;
  2: 1,  -4,    3;
  3: 1,  -9,   23,    -15;
  4: 1, -16,   86,   -176,    105;
  5: 1, -25,  230,   -950,   1689,     -945;
  6: 1, -36,  505,  -3480,  12139,   -19524,    10395;
  7: 1, -49,  973, -10045,  57379,  -177331,   264207,   -135135;
  8: 1, -64, 1708, -24640, 208054, -1038016,  2924172,  -4098240,  2027025;
  9: 1, -81, 2796, -53676, 626934, -4574934, 20570444, -53809164, 71697105, -34459425;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(2*n-1-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2*n - 1 - x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(2*n - 1 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-d7c9e146299b18aa
Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-d7c9e146299b18aa
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-d7c9e146299b18aa

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image affine, shift -1 e-d7c9e146299b18aa

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-59b849691f316c6f root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-ef36674c01230a86 root: theorem check
    
    Parent-row provers of polytool polytool ?

Used in 2 proof steps

e-557516b36262f3b9 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...…
e-557516b36262f3b9 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...…

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003