Real-rooted atlas

A033282 → x ↦ x+1 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(\left(-6 - 5\,n + 3\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-6 - 7\,n + 9\,n^{2} - 2\,n^{3}\right)\,P_{n-4}(x) + \left(48 + 12\,x + 40\,n + 10\,n x - 96\,n^{2} - 24\,n^{2} x + 32\,n^{3} + 8\,n^{3} x\right)\,P_{n-3}(x) + \left(-48 - 6\,x + 34\,n + 4\,n x + 186\,n^{2} + 24\,n^{2} x - 124\,n^{3} - 16\,n^{3} x\right)\,P_{n-2}(x) + \left(-24 - 6\,x - 56\,n - 14\,n x + 32\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)
\(P_{2}(x) = 31 + 30\,x + 5\,x^{2}\)
\(P_{3}(x) = 215 + 352\,x + 133\,x^{2} + 14\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
23130566
321535213314714
4159737802366546428331
5124253891835190131702178132102013
6999553916744727192522526669385804291292302
7824675389010859488554205410156885331946233605143016792398

All 15 rows

Technical details
Node id
n-29616fcd11ccaced · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-92f2350ecb06de83

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        5,         2;
  2:       31,        30,         5;
  3:      215,       352,       133,        14;
  4:     1597,      3780,      2366,       546,        42;
  5:    12425,     38918,     35190,     13170,      2178,       132;
  6:    99955,    391674,    472719,    252252,     66693,      8580,      429;
  7:   824675,   3890108,   5948855,   4205410,   1568853,    319462,    33605,    1430;
  8:  6939769,  38328840,  71566396,  63866660,  31327296,   8930350,  1476046,  131274,   4862;
  9: 59334605, 375784138, 833321164, 907052692, 558693408, 208118586, 47866390, 6651658, 512278, 16796;
%F Formula
The row polynomials satisfy (2*n^3+3*n^2-5*n-6)*P(n,x) = -(2*n^3-9*n^2+7*n+6)*P(n-4,x) + (32*n^3-96*n^2+40*n+48+(8*n^3-24*n^2+10*n+12)*x)*P(n-3,x) - (124*n^3-186*n^2-34*n+48+(16*n^3-24*n^2-4*n+6)*x)*P(n-2,x) + (32*n^3-56*n-24+(8*n^3-14*n-6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 31 + 30*x + 5*x^2, P(3,x) = 215 + 352*x + 133*x^2 + 14*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[2, x_] := 31 + 30*x + 5*x^2;
P[3, x_] := 215 + 352*x + 133*x^2 + 14*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 9*n^2 + 7*n + 6)*P[n - 4, x] + (32*n^3 - 96*n^2 + 40*n + 48 + (8*n^3 - 24*n^2 + 10*n + 12)*x)*P[n - 3, x] - (124*n^3 - 186*n^2 - 34*n + 48 + (16*n^3 - 24*n^2 - 4*n + 6)*x)*P[n - 2, x] + (32*n^3 - 56*n - 24 + (8*n^3 - 14*n - 6)*x)*P[n - 1, x])/(2*n^3 + 3*n^2 - 5*n - 6)]];
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] = 5 + 2*x; v[3] = 31 + 30*x + 5*x^2; v[4] = 215 + 352*x + 133*x^2 + 14*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(2*n^3 - 9*n^2 + 7*n + 6)*v[i - 4] + (32*n^3 - 96*n^2 + 40*n + 48 + (8*n^3 - 24*n^2 + 10*n + 12)*x)*v[i - 3] - (124*n^3 - 186*n^2 - 34*n + 48 + (16*n^3 - 24*n^2 - 4*n + 6)*x)*v[i - 2] + (32*n^3 - 56*n - 24 + (8*n^3 - 14*n - 6)*x)*v[i - 1])/(2*n^3 + 3*n^2 - 5*n - 6)); 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)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

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