Real-rooted atlas

A179900: Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.

A179900

Proved for all n: all roots real, in (3, 7) · interlacing · alternating coefficient signs what this means

Recurrence from OEIS A179900
\(P_{n}(x) = \left(5 - x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 - x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15-14
224-10115
3115-7315-156
4551-470147-201209
52640-28281190-24625-1780
612649-163108631-2400370-3012911
760605-9137158275-203854225-51935-110864

Show all 15 stored rows

Generated b-files: A010905 (row sums), A179900, A195503 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-490435c99149594f · revision 1 · JSON
Definition source
oeis_catalog:A179900
Fingerprint
f6-68cdd9b43eb15520

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 0)
  • 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): no

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       5,       -1;
  2:      24,      -10,       1;
  3:     115,      -73,      15,       -1;
  4:     551,     -470,     147,      -20,      1;
  5:    2640,    -2828,    1190,     -246,     25,     -1;
  6:   12649,   -16310,    8631,    -2400,    370,    -30,     1;
  7:   60605,   -91371,   58275,   -20385,   4225,   -519,    35,   -1;
  8:  290376,  -501150,  374115,  -157800,  41140,  -6790,   693,  -40,  1;
  9: 1391275, -2704755, 2313450, -1142730, 359275, -74571, 10220, -892, 45, -1;
%F Formula
The row polynomials satisfy P(n,x) = (5-x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 - x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 - x;
P[n_, x_] := P[n, x] = Expand[(5 - x)*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 - x; for(n = 2, N, my(i = n + 1); v[i] = (5 - x)*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A179900: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,       -1;
  2:      24,      -10,       1;
  3:     115,      -73,      15,       -1;
  4:     551,     -470,     147,      -20,      1;
  5:    2640,    -2828,    1190,     -246,     25,     -1;
  6:   12649,   -16310,    8631,    -2400,    370,    -30,     1;
  7:   60605,   -91371,   58275,   -20385,   4225,   -519,    35,   -1;
  8:  290376,  -501150,  374115,  -157800,  41140,  -6790,   693,  -40,  1;
  9: 1391275, -2704755, 2313450, -1142730, 359275, -74571, 10220, -892, 45, -1;
%F Formula
The row polynomials satisfy P(n,x) = (5-x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 - x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 - x;
P[n_, x_] := P[n, x] = Expand[(5 - x)*P[n - 1, x] - P[n - 2, 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; v[2] = 5 - x; for(n = 2, N, my(i = n + 1); v[i] = (5 - x)*v[i - 1] - v[i - 2]); 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b04f6384e5144344
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-19b03bdc222aebaf, e-b04f6384e5144344
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (3, 7)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-19b03bdc222aebaf
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-19b03bdc222aebaf

Proof chains

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

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023