Real-rooted atlas

A133214 → basis A008277

nonnegative coefficients

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

Recurrence
\(\left(2 + n\right)\,P_{n}(x) = \left(-1 - 8\,x - 4\,x^{2} - n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-8\,x - 8\,x^{2} - 8\,n x - 8\,n x^{2}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - 4\,n x^{2}\right)\,P_{n-2}''(x) + \left(9 + 6\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(6\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 20\,x + 4\,x^{2}\)
\(P_{1}(x) = 63 + 182\,x + 84\,x^{2} + 8\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01320437
163182848337
232116001312288163537
31683138421808068808803241397
4898911891623304413648029840249664529829
54863910192862888228243840880472011401667201287320145
62657298738816349361924081728018989376401049639782417408256108173377
7146256375036962416095104654429056411187392120672384176870401296384437765121697911173

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-bf4beaa81304b08d · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-d6c6efc0e156b65d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:       13,         20,           4;
  1:       63,        182,          84,            8;
  2:      321,       1600,        1312,          288,           16;
  3:     1683,      13842,       18080,         6880,          880,          32;
  4:     8989,     118916,      233044,       136480,        29840,        2496,          64;
  5:    48639,    1019286,     2888228,      2438408,       804720,      114016,        6720,        128;
  6:   265729,    8738816,    34936192,     40817280,     18989376,     4010496,      397824,      17408,       256;
  7:  1462563,   75036962,   416095104,    654429056,    411187392,   120672384,    17687040,    1296384,     43776,      512;
  8:  8097453,  645711220,  4906345444,  10185774720,   8395013440,  3275817216,   660511488,   71070720,   4005120,   107520,   1024;
  9: 45046719, 5569949046, 57478587188, 155240308168, 164357068480, 82747541888, 21944009472, 3224111616, 265393920, 11855360, 259072, 2048;
%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+9+(4*n+6)*x)*P(n-1,x) + (4*n+6)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 13 + 20*x + 4*x^2, P(1,x) = 63 + 182*x + 84*x^2 + 8*x^3.
%t Mathematica
P[0, x_] := 13 + 20*x + 4*x^2;
P[1, x_] := 63 + 182*x + 84*x^2 + 8*x^3;
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 + 9 + (4*n + 6)*x)*P[n - 1, x] + (4*n + 6)*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] = 13 + 20*x + 4*x^2; v[2] = 63 + 182*x + 84*x^2 + 8*x^3; 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 + 9 + (4*n + 6)*x)*v[i - 1] + (4*n + 6)*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-033100947c42f87a
  • 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-033100947c42f87a
  • 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-033100947c42f87a

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