Real-rooted atlas

A114687 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(2 + n\right)\,P_{n}(x) = \left(1 + 8\,x + 4\,x^{2} - n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(8\,x + 8\,x^{2} - 8\,n x - 8\,n x^{2}\right)\,P_{n-2}'(x) + \left(4\,x^{2} - 4\,n x^{2}\right)\,P_{n-2}''(x) + \left(3 + 2\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21118433
345140728265
41971060960240162473
590380221148048807203226037
64279611661305648428020720201664303089
720793471032144782413349284872007795253761283845233

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-58667740890593d7 · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-d95c25699e682aba

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 ?

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:      1;
  1:      3,        2;
  2:     11,       18,         4;
  3:     45,      140,        72,         8;
  4:    197,     1060,       960,       240,        16;
  5:    903,     8022,     11480,      4880,       720,       32;
  6:   4279,    61166,    130564,     84280,     20720,     2016,       64;
  7:  20793,   471032,   1447824,   1334928,    487200,    77952,     5376,    128;
  8: 103049,  3665160,  15848704,  20091456,  10221120,  2392320,   268800,  13824,   256;
  9: 518859, 28805546, 172421040, 292867520, 199628352, 64111488, 10429440, 867840, 34560, 512;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(n-1+(8*n-8)*x+(4*n-4)*x^2)*P(n-2,x) - ((8*n-8)*x+(8*n-8)*x^2)*d/dx P(n-2,x) - (4*n-4)*x^2*d^2/dx^2 P(n-2,x) + (6*n+3+(4*n+2)*x)*P(n-1,x) + (4*n+2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (8*n - 8)*x + (4*n - 4)*x^2)*P[n - 2, x] - ((8*n - 8)*x + (8*n - 8)*x^2)*D[P[n - 2, x], x] - (4*n - 4)*x^2*D[P[n - 2, x], {x, 2}] + (6*n + 3 + (4*n + 2)*x)*P[n - 1, x] + (4*n + 2)*x*D[P[n - 1, x], x])/(n + 2)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (8*n - 8)*x + (4*n - 4)*x^2)*v[i - 2] - ((8*n - 8)*x + (8*n - 8)*x^2)*deriv(v[i - 2], x) - (4*n - 4)*x^2*deriv(deriv(v[i - 2], x), x) + (6*n + 3 + (4*n + 2)*x)*v[i - 1] + (4*n + 2)*x*deriv(v[i - 1], x))/(n + 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.

Proved properties

  • 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-9b2f05e35e68edbc
  • 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-9b2f05e35e68edbc
  • 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-9b2f05e35e68edbc

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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023