Real-rooted atlas

A125662 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
248519108
3329651270271277
422557446545412428116478
5154568057492970362075265243230715
61059378446621444878856980208575211417293482902
7726103868385721224430180101076528600109058481648218756347516
84976784881722173005242833509780221778651464355750753122233061806561971698923
93411138588796866541501079006499211094442584324014907700082633964482450752211219311968317777057876
1023380291188938485885631478441211608672318210341078969646030912389110519090011476093780108295866403501559049343611253889
11160250899288751925948754623635220201983271305423086393305301321421671638414127065744744618453217784331255462058425142898581771476991721553132
12109837600338834329378681002004825129234465107591342498202346295363595116385523714306389749513332583677244046459150900239058945200191484097249955454531441149288813329330
137528381123987785287328311321563024451805794820335022941047671141404680939064887032184465099108237792135968137031538243720627651272721316605381188011661706774256392917271832515943233335584730106529
145160029186408712770582415917343474697345419633716335780902215943775525662662376648977506685114430365977291839519907932482400011629737653045491656018939331351511846645829387368758682430654048321591493833478296977790619207615122
Technical details
Node id
n-2f336a34b657c54a · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-7eeb2fb13d39a5eb

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:        7,         3;
  2:       48,        51,          9;
  3:      329,       651,        270,         27;
  4:     2255,      7446,       5454,       1242,         81;
  5:    15456,     80574,      92970,      36207,       5265,        243;
  6:   105937,    844662,    1444878,     856980,     208575,      21141,       729;
  7:   726103,   8683857,   21224430,   18010107,    6528600,    1090584,     81648,     2187;
  8:  4976784,  88172217,  300524283,  350978022,  177865146,   43557507,   5312223,   306180,    6561;
  9: 34111385, 887968665, 4150107900, 6499211094, 4425843240, 1490770008, 263396448, 24507522, 1121931, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 3*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 3*x)*P[n - 1, x] + 3*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] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 3*x)*v[i - 1] + 3*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