Real-rooted atlas

A207823 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 12\,x - 6\,n - 18\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n}(x) = \left(-x + 24\,x^{2} + 72\,x^{3} - 2\,n - 37\,n x - 132\,n x^{2} - 108\,n x^{3} + 4\,n^{2} + 30\,n^{2} x + 60\,n^{2} x^{2} + 36\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 24\,x^{3} + 36\,x^{4} - 2\,n x - 19\,n x^{2} - 48\,n x^{3} - 36\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
283038
31292168272
4161866968361734
52031217764364390010372
624470360013464245601746659584
7286606360320408892012880876048332864

All 15 rows

Technical details
Node id
n-7d50533e454457d7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2ea1a9b89b42c8bc

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:  4;
  2:  8,   30;
  3: 12,   92,   168;
  4: 16,  186,   696,    836;
  5: 20,  312,  1776,   4364,   3900;
  6: 24,  470,  3600,  13464,  24560,   17466;
  7: 28,  660,  6360,  32040,  88920,  128808,   76048;
  8: 32,  882, 10248,  65020, 243120,  535428,  642768,  324360;
  9: 36, 1136, 15456, 118356, 557620, 1654344, 3016944, 3090984, 1361844;
%F Formula
The row polynomials satisfy (4*n^2-6*n+2+(6*n^2-18*n+12)*x)*P(n,x) = (4*n^2-2*n+(30*n^2-37*n-1)*x+(60*n^2-132*n+24)*x^2+(36*n^2-108*n+72)*x^3)*P(n-1,x) - (2*n*x+(19*n-3)*x^2+(48*n-24)*x^3+(36*n-36)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (30*n^2 - 37*n - 1)*x + (60*n^2 - 132*n + 24)*x^2 + (36*n^2 - 108*n + 72)*x^3)*P[n - 1, x] - (2*n*x + (19*n - 3)*x^2 + (48*n - 24)*x^3 + (36*n - 36)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (6*n^2 - 18*n + 12)*x)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n + (30*n^2 - 37*n - 1)*x + (60*n^2 - 132*n + 24)*x^2 + (36*n^2 - 108*n + 72)*x^3)*v[i - 1] - (2*n*x + (19*n - 3)*x^2 + (48*n - 24)*x^3 + (36*n - 36)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (6*n^2 - 18*n + 12)*x)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023