Real-rooted atlas

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 ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-110
33-410
4-1523-910
5105-17686-1610
6-9451689-950230-2510
710395-1952412139-3480505-3610
8-135135264207-17733157379-10045973-4910
92027025-40982402924172-1038016208054-246401708-6410
10-3445942571697105-5380916420570444-4574934626934-536762796-8110
11654729075-13967044201094071221-444647600107494190-164866801646778-1068004335-10010
12-1374931057529985521895-2437220006110431670821-2702025590453714470-510690183889578-1978356435-12110
13316234143225-703416314160590546123298-26430062894472578259391-131374584001628301884-1405293128439783-3458409218-14410
14-790585358062517901641997225-154670693966107198061846898-2078757113719401014719391-538450055005141534684-35152388717085783-57629012818-16910
15213458046676875-491250187505700435512515705695-20981473926285663324503917311-129061545372761854829867891-19266644196814632679633-81284002832645613-92237617381-19610

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 ?

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

Used in 2 proof steps

No candidate equalities or related sequences.

References

  1. [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