Real-rooted atlas

A207823 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(9 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19413
2808816184
37111460624642859
463192173616432384025648583
556160306384365664146304217601024897296
64991214185896742539245312001128960116736409617891401
744359295610674814260958412471046446327040790016060211216384382708421

All 15 rows

Technical details
Node id
n-215c9aaad9ec9923 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-054f2160f6497013

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:         9,          4;
  2:        80,         88,          16;
  3:       711,       1460,         624,          64;
  4:      6319,      21736,       16432,        3840,         256;
  5:     56160,     306384,      365664,      146304,       21760,        1024;
  6:    499121,    4185896,     7425392,     4531200,     1128960,      116736,       4096;
  7:   4435929,   56106748,   142609584,   124710464,    46327040,     7900160,     602112,     16384;
  8:  39424240,  742945544,  2641364528,  3184826880,  1655888896,   414296064,   51466240,   3014656,    65536;
  9: 350382231, 9759882284, 47732369568, 77322112128, 54090202880, 18630241280, 3354968064, 317390848, 14745600, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (9+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 4*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (9 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
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] = 9 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (9 + 4*x)*v[i - 1] + 4*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 - 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. [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