Real-rooted atlas

A111125 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
27613
3417836155
42397086842161847
51393560483165400129622009
68119412748283682080401767776262261
747321290754736344984096732240287712466563125123
82758071987176607975210240560102578406111936200620827993637239215

All 15 rows

Technical details
Node id
n-37b72160314f4b8f · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-93cb1c672285184d

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:       7,        6;
   3:      41,       78,        36;
   4:     239,      708,       684,       216;
   5:    1393,     5604,      8316,      5400,       1296;
   6:    8119,    41274,     82836,     82080,      40176,       7776;
   7:   47321,   290754,    736344,    984096,     732240,     287712,     46656;
   8:  275807,  1987176,   6079752,  10240560,   10257840,    6111936,   2006208,    279936;
   9: 1607521, 13287144,  47665224,  96937776,  122258160,   97930944,  48662208,  13716864,  1679616;
  10: 9369319, 87380814, 359634456, 857377440, 1304917776, 1315022688, 877552704, 373994496, 92378880, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 7 + 6*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 6*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] = 7 + 6*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (6 + 6*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