Real-rooted atlas

A207606 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 89 + 140\,x + 32\,x^{2}\)
\(P_{1}(x) = 610 + 1888\,x + 1040\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
08914032261
1610188810401283666
241812306823120659251257473
328657268584438032217600381442048993065
41964183045984762169658799361747200207872819218707298
51346269340229121260713441419879686366720012599296108544032768380813197
692274653765911242019540112319617683220305502725946419208403763254886401310728316385069
763245986414538931231673394832686635322885942369638424164933632498264883252822016027066368524288193672652082

All 15 rows

Technical details
Node id
n-29c226511ecc97cc · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5802c90b5b83bab2

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         89,          140,            32;
  1:        610,         1888,          1040,            128;
  2:       4181,        23068,         23120,           6592,            512;
  3:      28657,       268584,        438032,         217600,          38144,           2048;
  4:     196418,      3045984,       7621696,        5879936,        1747200,         207872,           8192;
  5:    1346269,     34022912,     126071344,      141987968,       63667200,       12599296,        1085440,         32768;
  6:    9227465,    376591124,    2019540112,     3196176832,     2030550272,      594641920,       84037632,       5488640,       131072;
  7:   63245986,   4145389312,   31673394832,    68663532288,    59423696384,    24164933632,     4982648832,     528220160,     27066368,      524288;
  8:  433494437,  45475675252,  489662939616,  1428104515968,  1639368595712,   889553351680,   251037810688,   38412812288,   3168337920,   130809856,   2097152;
  9: 2971215073, 497821016208, 7495173400416, 29023974029568, 43358472068864, 30551249944576, 11335402889216, 2348071452672, 277189361664, 18297651200, 621805568, 8388608;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 89 + 140*x + 32*x^2, P(1,x) = 610 + 1888*x + 1040*x^2 + 128*x^3.
%t Mathematica
P[0, x_] := 89 + 140*x + 32*x^2;
P[1, x_] := 610 + 1888*x + 1040*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 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] = 89 + 140*x + 32*x^2; v[2] = 610 + 1888*x + 1040*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 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