Real-rooted atlas

A008459 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(16 + 48\,x + 16\,x^{2} - 16\,n - 48\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(48\,x + 32\,x^{2} - 48\,n x - 32\,n x^{2}\right)\,P_{n-2}'(x) + \left(16\,x^{2} - 16\,n x^{2}\right)\,P_{n-2}''(x) + \left(-6 - 4\,x + 12\,n + 8\,n x\right)\,P_{n-1}(x) + \left(-4\,x + 8\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2467216134
33961144576642180
436061768015520384025640902
533876270424373760156800230401024858924
63245564119024848713654118401326080129024409619801756
731518966267870418612339216996134462003200994918468812816384494572232

All 15 rows

Technical details
Node id
n-2098ed19ade7836f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5d48e54e6c6fa559

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:         6,           4;
  2:        46,          72,          16;
  3:       396,        1144,         576,           64;
  4:      3606,       17680,       15520,         3840,         256;
  5:     33876,      270424,      373760,       156800,       23040,        1024;
  6:    324556,     4119024,     8487136,      5411840,     1326080,      129024,       4096;
  7:   3151896,    62678704,   186123392,    169961344,    62003200,     9949184,     688128,     16384;
  8:  30915046,   954568224,  3993574720,   5030462976,  2567502336,   605036544,   68468736,   3538944,    65536;
  9: 305543556, 14564110744, 84494384640, 143112720640, 98254205952, 31901657088, 5250416640, 441384960, 17694720, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = -(16*n-16+(48*n-48)*x+(16*n-16)*x^2)*P(n-2,x) - ((48*n-48)*x+(32*n-32)*x^2)*d/dx P(n-2,x) - (16*n-16)*x^2*d^2/dx^2 P(n-2,x) + (12*n-6+(8*n-4)*x)*P(n-1,x) + (8*n-4)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n - 16 + (48*n - 48)*x + (16*n - 16)*x^2)*P[n - 2, x] - ((48*n - 48)*x + (32*n - 32)*x^2)*D[P[n - 2, x], x] - (16*n - 16)*x^2*D[P[n - 2, x], {x, 2}] + (12*n - 6 + (8*n - 4)*x)*P[n - 1, x] + (8*n - 4)*x*D[P[n - 1, x], x])/(n)]];
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] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n - 16 + (48*n - 48)*x + (16*n - 16)*x^2)*v[i - 2] - ((48*n - 48)*x + (32*n - 32)*x^2)*deriv(v[i - 2], x) - (16*n - 16)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n - 6 + (8*n - 4)*x)*v[i - 1] + (8*n - 4)*x*deriv(v[i - 1], x))/(n)); 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 (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  6. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023