Real-rooted atlas

A001263 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n + n^{2}\right)\,P_{n}(x) = \left(-6\,x + 2\,n - 3\,n x + 4\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x) + \left(-x - 3\,x^{2} - 2\,n x - 6\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2151833
36316281306
4252100811343242718
5990540097206480121523805
63861267306682572900328054374207495
715015126126405405623700467775153090153091806420

All 15 rows

Technical details
Node id
n-62169c7bee8719c8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dae6101d606a19f7

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      3;
  2:     15,      18;
  3:     63,     162,       81;
  4:    252,    1008,     1134,      324;
  5:    990,    5400,     9720,     6480,     1215;
  6:   3861,   26730,    66825,    72900,    32805,     4374;
  7:  15015,  126126,   405405,   623700,   467775,   153090,    15309;
  8:  58344,  576576,  2270268,  4540536,  4864860,  2694384,   673596,   52488;
  9: 226746, 2577744, 12029472, 29719872, 41793570, 33434856, 14329224, 2834352, 177147;
%F Formula
The row polynomials satisfy (n^2+n-2)*P(n,x) = (4*n^2+2*n+(9*n^2-3*n-6)*x)*P(n-1,x) - ((2*n+1)*x+(6*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (9*n^2 - 3*n - 6)*x)*P[n - 1, x] - ((2*n + 1)*x + (6*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n + (9*n^2 - 3*n - 6)*x)*v[i - 1] - ((2*n + 1)*x + (6*n + 3)*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 - 1, v[i])))};
T(9)

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