Real-rooted atlas

A145597 → γ → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 4\,n + n^{2}\right)\,P_{n}(x) = \left(12 - 24\,x + 8\,n - 4\,n x + n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - 16\,x^{2} + 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2Pn(1)
00
10
233
31515
4451863
5105126231
621050484798
737815127562646

All 15 rows

Technical details
Node id
n-3347371a4f01b6c3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d60dbc403e0c506a

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   0;
  2:   3;
  3:  15;
  4:  45,   18;
  5: 105,  126;
  6: 210,  504,    84;
  7: 378, 1512,   756;
  8: 630, 3780,  3780,  360;
  9: 990, 8316, 13860, 3960;
%F Formula
The row polynomials satisfy (n^2+4*n-12)*P(n,x) = (n^2+8*n+12+(4*n^2-4*n-24)*x)*P(n-1,x) + ((2*n+4)*x-(8*n+16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 8*n + 12 + (4*n^2 - 4*n - 24)*x)*P[n - 1, x] + ((2*n + 4)*x - (8*n + 16)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n - 12)]];
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] = 0; v[3] = 3; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + 8*n + 12 + (4*n^2 - 4*n - 24)*x)*v[i - 1] + ((2*n + 4)*x - (8*n + 16)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n - 12)); 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

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