Real-rooted atlas

A114656 → basis A008277

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3119121
44567181131
5197475225301928
6903332124405654517275
7427923177246618680119063162051
820793162247239694119301245702226841568916

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-505fa69e524feade · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-a2e2bf0142c1e4e9

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,       1;
   3:     11,       9,        1;
   4:     45,      67,       18,        1;
   5:    197,     475,      225,       30,       1;
   6:    903,    3321,     2440,      565,      45,       1;
   7:   4279,   23177,    24661,     8680,    1190,      63,      1;
   8:  20793,  162247,   239694,   119301,   24570,    2226,     84,    1;
   9: 103049, 1141623,  2276737,  1532034,  440055,   59430,   3822,  108,   1;
  10: 518859, 8080693, 21327180, 18821965, 7197939, 1343559, 128100, 6150, 135, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(n-2-(n-2)*x+(n-2)*x^2)*P(n-2,x) + ((n-2)*x-(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (6*n-3+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 - (n - 2)*x + (n - 2)*x^2)*P[n - 2, x] + ((n - 2)*x - (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (6*n - 3 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x])/(n + 1)]];
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 + x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 - (n - 2)*x + (n - 2)*x^2)*v[i - 2] + ((n - 2)*x - (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (6*n - 3 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x))/(n + 1)); 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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-262d5bb70cb5d431
  • 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-262d5bb70cb5d431
  • 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-262d5bb70cb5d431

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