Real-rooted atlas

A110441: Triangular array formed by the Mersenne numbers.

A110441 nonnegative coefficients

Proved for all n: all roots real, in (−6, 0) · interlacing · coefficients of one sign · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A110441
\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(-2\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
276114
315239148
4317248121164
56320119882151560
61275226994201251811912
72551291222317957651772116528
85113084656267683840126023824122288
9102371811832423276167587266193230827176096
1020471639849029746166587036036125862808387301259808
1140953687912683722632523871015944669930203943915475331887040
128191819363193326565808107156449763440641254963136552805723613028544
1316383180241786258183642226113052426751153730867976421176146255693067839110340096
1432767393234190035149823648048907860160663505473325608125231733996665901889279342135303296

Generated b-files: A007070 (row sums), A110441 ?

Computed in your browser ?

Technical details
Node id
n-9b5c7e97e5191da6 · revision 1 · JSON
Definition source
oeis_catalog:A110441
Fingerprint
f6-37c3559a5d3b5b73

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): yes
  • 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:    3,    1;
  2:    7,    6,     1;
  3:   15,   23,     9,     1;
  4:   31,   72,    48,    12,     1;
  5:   63,  201,   198,    82,    15,    1;
  6:  127,  522,   699,   420,   125,   18,    1;
  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;
  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) - 2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] - 2*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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1] - 2*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.

A110441: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    3,    1;
  2:    7,    6,     1;
  3:   15,   23,     9,     1;
  4:   31,   72,    48,    12,     1;
  5:   63,  201,   198,    82,    15,    1;
  6:  127,  522,   699,   420,   125,   18,    1;
  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;
  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) - 2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] - 2*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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1] - 2*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-7eb461f112fb8b45
  • 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-02d1e2cf79c791bd, e-7eb461f112fb8b45, e-c01294f1353c0463
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-c01294f1353c0463
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n 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-02d1e2cf79c791bd, e-c01294f1353c0463
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-02d1e2cf79c791bd, e-c01294f1353c0463
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-02d1e2cf79c791bd
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-sqrt(2), sqrt(2)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has a real-rooted image
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-4036622c95205ccf
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-sqrt(2), sqrt(2)] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-4036622c95205ccf
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-sqrt(2), 0] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every PF input with all roots in [-sqrt(2), 0] has a PF image; no larger [-R, 0] works
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-4036622c95205ccf

Proof chains

4 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

Used in 14 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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