Real-rooted atlas

A123519 → basis A008288

nonnegative coefficients

Proved for all n: all roots real, in [−3−2√2, −3+2√2] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(-1\right)\,P_{n-4}(x) + \left(6 + 2\,x\right)\,P_{n-3}(x) + \left(-10\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 11 + 18\,x + 4\,x^{2}\)
\(P_{1}(x) = 41 + 112\,x + 60\,x^{2} + 8\,x^{3}\)
\(P_{2}(x) = 153 + 592\,x + 548\,x^{2} + 168\,x^{3} + 16\,x^{4}\)
\(P_{3}(x) = 571 + 2866\,x + 3932\,x^{2} + 2032\,x^{3} + 432\,x^{4} + 32\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01118433
141112608221
2153592548168161477
3571286639322032432329865
42131131542442418544651210566465885
579535827213794414188872272190722496128440025
6296812517447287449616646589282494725254457602562938793
7110771106742636654325966672522990426432967917441384961305651219627309

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-1c7cd5e31010a2aa · revision 2 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-d987e0954c3bd72b

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 ?

No OEIS matches.

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:      11,       18,        4;
  1:      41,      112,       60,         8;
  2:     153,      592,      548,       168,        16;
  3:     571,     2866,     3932,      2032,       432,        32;
  4:    2131,    13154,    24424,     18544,      6512,      1056,       64;
  5:    7953,    58272,   137944,    141888,     72272,     19072,     2496,      128;
  6:   29681,   251744,   728744,    961664,    658928,    249472,    52544,     5760,     256;
  7:  110771,  1067426,  3665432,   5966672,   5229904,   2643296,   791744,   138496,   13056,    512;
  8:  413403,  4461042, 17759788,  34622928,  37434384,  24082784,  9564672,  2362624,  353024,  29184,  1024;
  9: 1542841, 18430352, 83564500, 190676008, 247357744, 195728128, 98447872, 32059648, 6725888, 876544, 64512, 2048;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-4,x) + (6+2*x)*P(n-3,x) - 10*P(n-2,x) + (6+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 11 + 18*x + 4*x^2, P(1,x) = 41 + 112*x + 60*x^2 + 8*x^3, P(2,x) = 153 + 592*x + 548*x^2 + 168*x^3 + 16*x^4, P(3,x) = 571 + 2866*x + 3932*x^2 + 2032*x^3 + 432*x^4 + 32*x^5.
%t Mathematica
P[0, x_] := 11 + 18*x + 4*x^2;
P[1, x_] := 41 + 112*x + 60*x^2 + 8*x^3;
P[2, x_] := 153 + 592*x + 548*x^2 + 168*x^3 + 16*x^4;
P[3, x_] := 571 + 2866*x + 3932*x^2 + 2032*x^3 + 432*x^4 + 32*x^5;
P[n_, x_] := P[n, x] = Expand[-P[n - 4, x] + (6 + 2*x)*P[n - 3, x] - 10*P[n - 2, x] + (6 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 18*x + 4*x^2; v[2] = 41 + 112*x + 60*x^2 + 8*x^3; v[3] = 153 + 592*x + 548*x^2 + 168*x^3 + 16*x^4; v[4] = 571 + 2866*x + 3932*x^2 + 2032*x^3 + 432*x^4 + 32*x^5; for(n = 4, N, my(i = n + 1); v[i] = -v[i - 4] + (6 + 2*x)*v[i - 3] - 10*v[i - 2] + (6 + 2*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 - 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 [-1564556493/268435456, -46056243/268435456]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b77c7252fe4ee4d8
  • 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-0686670e536a2f53, e-b77c7252fe4ee4d8
  • 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-0686670e536a2f53, e-b77c7252fe4ee4d8
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0686670e536a2f53

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