Real-rooted atlas

A114656 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x - 5\,n - 7\,n x + 5\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 + x^{2} + 2\,n - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
38210
45130384
5304292724672
61765238099014055280
71012817670105602580240641184
857799124026975453486057053787320320

All 15 rows

Technical details
Node id
n-f14e2bec4affca4e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-58fd21a6ca87df0d

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       0;
   2:       1;
   3:       8,       2;
   4:      51,      30,       3;
   5:     304,     292,      72,       4;
   6:    1765,    2380,     990,     140,       5;
   7:   10128,   17670,   10560,    2580,     240,      6;
   8:   57799,  124026,   97545,   34860,    5705,    378,     7;
   9:  329024,  839048,  822192,  392056,   95200,  11256,   560,   8;
  10: 1871145, 5532120, 6507900, 3908520, 1280790, 227304, 20412, 792, 9;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = (5*n^2-5*n+2+(3*n^2-7*n+2)*x)*P(n-1,x) + (2*n-1-(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 5*n + 2 + (3*n^2 - 7*n + 2)*x)*P[n - 1, x] + (2*n - 1 - (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((5*n^2 - 5*n + 2 + (3*n^2 - 7*n + 2)*x)*v[i - 1] + (2*n - 1 - (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n - 2)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023