Real-rooted atlas

A172431 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 24\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(7 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17815
2408848176
32176967362561905
41159485675365120128019951
56160317926480065024320006144205920
632689200304505024668928489216186368286722111201
717338312296723692832607360058905603354624103219213107221577935

All 15 rows

Technical details
Node id
n-37be3118d4ffb268 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e7d980e9e7184757

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,        8;
  2:      40,       88,        48;
  3:     217,      696,       736,       256;
  4:    1159,     4856,      7536,      5120,      1280;
  5:    6160,    31792,     64800,     65024,     32000,      6144;
  6:   32689,   200304,    505024,    668928,    489216,    186368,     28672;
  7:  173383,  1229672,   3692832,   6073600,   5890560,   3354624,   1032192,    131072;
  8:  919480,  7407496,  25811664,  50712064,  61285120,  46485504,  21504000,   5505024,   589824;
  9: 4875913, 43980072, 174419872, 398512640, 576825600, 546936832, 338362368, 130940928, 28508160, 2621440;
%F Formula
The row polynomials satisfy P(n,x) = -(9+24*x+16*x^2)*P(n-2,x) + (7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 24*x + 16*x^2)*P[n - 2, x] + (7 + 8*x)*P[n - 1, 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 24*x + 16*x^2)*v[i - 2] + (7 + 8*x)*v[i - 1]); 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

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023