Real-rooted atlas

A063007 → 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
\(\left(-14 + 25\,n - 13\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(36 - 45\,n + 17\,n^{2} - 2\,n^{3}\right)\,P_{n-4}(x) + \left(-252 - 126\,x + 324\,n + 162\,n x - 128\,n^{2} - 64\,n^{2} x + 16\,n^{3} + 8\,n^{3} x\right)\,P_{n-3}(x) + \left(370 - 210\,x - 498\,n + 284\,n x + 210\,n^{2} - 120\,n^{2} x - 28\,n^{3} + 16\,n^{3} x\right)\,P_{n-2}(x) + \left(-168 - 84\,x + 244\,n + 122\,n x - 112\,n^{2} - 56\,n^{2} x + 16\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)
\(P_{3}(x) = 13 + 24\,x + 6\,x^{2}\)
\(P_{4}(x) = 63 + 202\,x + 130\,x^{2} + 20\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31324643
46320213020415
532114801700630704201
61683101381750011270289825243741
78989668641569541512426598212936924463891
8486394306101289550170033410964583581165662834324983767

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-59aff779bb582ee4 · revision 2 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-313435422367fb28

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): 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,        2;
   3:      13,       24,        6;
   4:      63,      202,      130,        20;
   5:     321,     1480,     1700,       630,        70;
   6:    1683,    10138,    17500,     11270,      2898,       252;
   7:    8989,    66864,   156954,    151242,     65982,     12936,      924;
   8:   48639,   430610,  1289550,   1700334,   1096458,    358116,    56628,    3432;
   9:  265729,  2728464,  9981384,  16964772,  14902932,   7110246,  1847274,  244530,   12870;
  10: 1462563, 17086258, 74022520, 155439460, 176066748, 113649822, 42701230, 9187750, 1045330, 48620;
%F Formula
The row polynomials satisfy (2*n^3-13*n^2+25*n-14)*P(n,x) = -(2*n^3-17*n^2+45*n-36)*P(n-4,x) + (16*n^3-128*n^2+324*n-252+(8*n^3-64*n^2+162*n-126)*x)*P(n-3,x) - (28*n^3-210*n^2+498*n-370-(16*n^3-120*n^2+284*n-210)*x)*P(n-2,x) + (16*n^3-112*n^2+244*n-168+(8*n^3-56*n^2+122*n-84)*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 13 + 24*x + 6*x^2, P(4,x) = 63 + 202*x + 130*x^2 + 20*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 13 + 24*x + 6*x^2;
P[4, x_] := 63 + 202*x + 130*x^2 + 20*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 17*n^2 + 45*n - 36)*P[n - 4, x] + (16*n^3 - 128*n^2 + 324*n - 252 + (8*n^3 - 64*n^2 + 162*n - 126)*x)*P[n - 3, x] - (28*n^3 - 210*n^2 + 498*n - 370 - (16*n^3 - 120*n^2 + 284*n - 210)*x)*P[n - 2, x] + (16*n^3 - 112*n^2 + 244*n - 168 + (8*n^3 - 56*n^2 + 122*n - 84)*x)*P[n - 1, x])/(2*n^3 - 13*n^2 + 25*n - 14)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 13 + 24*x + 6*x^2; v[4] = 63 + 202*x + 130*x^2 + 20*x^3; for(n = 5, N, my(i = n); v[i] = (-(2*n^3 - 17*n^2 + 45*n - 36)*v[i - 4] + (16*n^3 - 128*n^2 + 324*n - 252 + (8*n^3 - 64*n^2 + 162*n - 126)*x)*v[i - 3] - (28*n^3 - 210*n^2 + 498*n - 370 - (16*n^3 - 120*n^2 + 284*n - 210)*x)*v[i - 2] + (16*n^3 - 112*n^2 + 244*n - 168 + (8*n^3 - 56*n^2 + 122*n - 84)*x)*v[i - 1])/(2*n^3 - 13*n^2 + 25*n - 14)); 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-0d826cc18bc3a95c
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-0d826cc18bc3a95c, e-422298c8a4851a11
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-0d826cc18bc3a95c, e-422298c8a4851a11
  • Real-rooted. Every row P_n, n >= 1, 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-422298c8a4851a11

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