Real-rooted atlas

A094816 → basis A013611 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(4 + 4\,x + n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
3415216109
4316572336641288
5272162565264192025616417
625946711567632040000102401024224686
727180185522010978247356802713605222440963288205
83105936109229081613684812979456613760017059842580481638451263164

All 15 rows

Technical details
Node id
n-ed40b6d46319efeb · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3633f7d4fe1210c3

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         1;
   2:         6,          4;
   3:        41,         52,         16;
   4:       316,        572,        336,         64;
   5:      2721,       6256,       5264,       1920,        256;
   6:     25946,      71156,      76320,      40000,      10240,       1024;
   7:    271801,     855220,    1097824,     735680,     271360,      52224,      4096;
   8:   3105936,   10922908,   16136848,   12979456,    6137600,    1705984,    258048,    16384;
   9:  38474561,  148435008,  245785888,  228130560,  129807104,   46362624,  10149888,  1245184,   65536;
  10: 513796366, 2144605092, 3905647680, 4073135744, 2680720896, 1154657280, 325484544, 57901056, 5898240, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*P[n - 2, x] + (n + 4 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 4 + 4*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, v[i])))};
T(10)

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