Real-rooted atlas

A078812 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,n\right)\,P_{n}(x) = \left(6 + 4\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(5 + 12\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
312820
4507224146
518040828864940
6600187220649601605656
719087608115208320288038432620
85878285845532054720292808064896182742

All 15 rows

Technical details
Node id
n-9716eddae19d8e20 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-63c89f2338138bfe

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:     2;
   3:    12,      8;
   4:    50,     72,     24;
   5:   180,    408,    288,      64;
   6:   600,   1872,   2064,     960,     160;
   7:  1908,   7608,  11520,    8320,    2880,    384;
   8:  5878,  28584,  55320,   54720,   29280,   8064,    896;
   9: 17700, 101656, 240192,  303360,  221760,  94080,  21504,  2048;
  10: 52390, 347184, 970224, 1495872, 1394400, 806400, 283136, 55296, 4608;
%F Formula
The row polynomials satisfy (4*n-8)*P(n,x) = (6*n+6+(4*n+4)*x)*P(n-1,x) + (5+12*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 6 + (4*n + 4)*x)*P[n - 1, x] + (5 + 12*x + 4*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((6*n + 6 + (4*n + 4)*x)*v[i - 1] + (5 + 12*x + 4*x^2)*deriv(v[i - 1], x))/(4*n - 8)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023