Real-rooted atlas

A126216 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2117119
34542121100
4197242104181562
590313757902102513304
6427977735595201537533120071
7207934387637968172904410616421124996
8103049247684250432138040448848680952521793774
95188591399293161895610481524150441030681579214046315120632
10264672379139551031164576710303584700109006821588027030199575133463102
111364886944812878649354205458579529421150106038242584428420750440552750881221060008
12710393732540647264052735283799306502321149059682714828081416565210877368568970369610211473830308
13372693519144213189925112255222597702250177447312984144789328335793268113188115757071355783104390486211719904186192
1419688015198195232633154687934311750291151713221767559702705303326971294357550550151537686152244742522812791535176279133167015401391

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 7\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-83e32d0bc1c10d45 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-0bcf293e4d350535)
Fingerprint
f6-cb8a225300b7095e

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 ?

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:     11,       7,       1;
  3:     45,      42,      12,       1;
  4:    197,     242,     104,      18,      1;
  5:    903,    1375,     790,     210,     25,      1;
  6:   4279,    7773,    5595,    2015,    375,     33,     1;
  7:  20793,   43876,   37968,   17290,   4410,    616,    42,    1;
  8: 103049,  247684,  250432,  138040,  44884,   8680,   952,   52,  1;
  9: 518859, 1399293, 1618956, 1048152, 415044, 103068, 15792, 1404, 63, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(n-1+(2*n-2)*x+(n-1)*x^2)*P(n-2,x) + (6*n+3+(2*n+1)*x)*P(n-1,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[Together[(-(n - 1 + (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x] + (6*n + 3 + (2*n + 1)*x)*P[n - 1, 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 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (2*n - 2)*x + (n - 1)*x^2)*v[i - 2] + (6*n + 3 + (2*n + 1)*x)*v[i - 1])/(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-3c8d43f1dfa996d6
  • 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-3c8d43f1dfa996d6, e-556b2ae7482db11d
  • 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-3c8d43f1dfa996d6, e-556b2ae7482db11d
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-3c8d43f1dfa996d6, e-556b2ae7482db11d
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-556b2ae7482db11d

Proof chains

3 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

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