Real-rooted atlas

A069466 → basis A008277

nonnegative coefficients

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

Recurrence
\(\left(n^{2}\right)\,P_{n}(x) = \left(-12\,x - 12\,x^{2} + 32\,n x + 32\,n x^{2} - 16\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x - 24\,x^{2} + 32\,n x + 64\,n x^{2} - 16\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(-12\,x^{2} + 32\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(4 + 2\,x - 16\,n - 8\,n x + 16\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
23642684
3400860300201580
44900175701099018207035350
56350436061235658010458010080252895608
685377674520601089488449341608362205266892425024692
7117786241551504243219936722093211125405400059579522642643432758523480

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-045ced6e64494a1f · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b334f8e538f872f3

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:          4,           2;
  2:         36,          42,            6;
  3:        400,         860,          300,           20;
  4:       4900,       17570,        10990,         1820,           70;
  5:      63504,      360612,       356580,       104580,        10080,         252;
  6:     853776,     7452060,     10894884,      4934160,       836220,       52668,        924;
  7:   11778624,   155150424,    321993672,    209321112,     54054000,     5957952,     264264,      3432;
  8:  165636900,  3254295330,   9332976510,   8327018700,   3035402370,   505404900,   39099060,   1287000,   12870;
  9: 2363904400, 68745908660, 267386613780, 317747598740, 155983267440, 36267651420, 4212728520, 241252440, 6126120, 48620;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -((16*n^2-32*n+12)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) - ((16*n^2-32*n+12)*x+(32*n^2-64*n+24)*x^2)*d/dx P(n-2,x) - (16*n^2-32*n+12)*x^2*d^2/dx^2 P(n-2,x) + (16*n^2-16*n+4+(8*n^2-8*n+2)*x)*P(n-1,x) + (8*n^2-8*n+2)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((16*n^2 - 32*n + 12)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x] - ((16*n^2 - 32*n + 12)*x + (32*n^2 - 64*n + 24)*x^2)*D[P[n - 2, x], x] - (16*n^2 - 32*n + 12)*x^2*D[P[n - 2, x], {x, 2}] + (16*n^2 - 16*n + 4 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] + (8*n^2 - 8*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-((16*n^2 - 32*n + 12)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2] - ((16*n^2 - 32*n + 12)*x + (32*n^2 - 64*n + 24)*x^2)*deriv(v[i - 2], x) - (16*n^2 - 32*n + 12)*x^2*deriv(deriv(v[i - 2], x), x) + (16*n^2 - 16*n + 4 + (8*n^2 - 8*n + 2)*x)*v[i - 1] + (8*n^2 - 8*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-62dc2e1e9139053e
  • 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-62dc2e1e9139053e
  • 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-62dc2e1e9139053e

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

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. [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