Real-rooted atlas

A086645 → basis A013611 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-16 - 48\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(-48\,x - 32\,x^{2}\right)\,P_{n-2}'(x) + \left(-16\,x^{2}\right)\,P_{n-2}''(x) + \left(12 + 8\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
3568016152
45761504672642816
560162688021248460825659008
6629764618245836802201602816010241357824
765945677312001473126486056961894400159744409633785856
86905856127320064352321536298532864998932481439334486016016384900243456

All 15 rows

Technical details
Node id
n-b6af2aa7abc12b78 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-057772c6c1047f32

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         1;
   2:         6,           4;
   3:        56,          80,           16;
   4:       576,        1504,          672,           64;
   5:      6016,       26880,        21248,         4608,          256;
   6:     62976,      461824,       583680,       220160,        28160,        1024;
   7:    659456,     7731200,     14731264,      8605696,      1894400,      159744,       4096;
   8:   6905856,   127320064,    352321536,    298532864,     99893248,    14393344,     860160,     16384;
   9:  72318976,  2075197440,   8133214208,   9589096448,   4550885376,   984481792,  100007936,   4456448,    65536;
  10: 757334016, 33602535424, 183301177344, 292351639552, 188368551936, 56951832576, 8609857536, 649592832, 22413312, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(16+48*x+16*x^2)*P(n-2,x) - (48*x+32*x^2)*d/dx P(n-2,x) - 16*x^2*d^2/dx^2 P(n-2,x) + (12+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 48*x + 16*x^2)*P[n - 2, x] - (48*x + 32*x^2)*D[P[n - 2, x], x] - 16*x^2*D[P[n - 2, x], {x, 2}] + (12 + 8*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; for(n = 3, N, my(i = n); v[i] = -(16 + 48*x + 16*x^2)*v[i - 2] - (48*x + 32*x^2)*deriv(v[i - 2], x) - 16*x^2*deriv(deriv(v[i - 2], x), x) + (12 + 8*x)*v[i - 1] + 8*x*deriv(v[i - 1], x)); 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)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [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
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023