Real-rooted atlas

A111125 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(60 - 6\,x^{2} - 88\,n + 7\,n x^{2} + 47\,n^{2} - 2\,n^{2} x^{2} - 13\,n^{3} + 2\,n^{4}\right)\,P_{n}(x) = \left(-8\,n - 4\,n x + 2\,n x^{2} + n x^{3} + 17\,n^{2} + 5\,n^{2} x - 3\,n^{2} x^{2} - n^{2} x^{3} - 3\,n^{3} - 2\,n^{3} x + 2\,n^{4} + n^{4} x\right)\,P_{n-1}(x) + \left(-24\,x + 2\,x^{2} + 6\,x^{3} + x^{4} + 32\,n x - 6\,n x^{3} - n x^{4} - 12\,n^{2} x - 3\,n^{2} x^{2} + 4\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3729
42820351
584108394235
6209418264645960
7455130012485209563624
8896345646502900900132712941
916328160145861258057801428175844349
102793175564012845448285951037421282249147255
11454335046995281426881168655762417248302427910476855
12708465780226941400660412160262752106260270484140340111513176
1310660117260482625102700012920001028160534660183000405005500407124721784
1415561200070967824243984036765903558168229824010069922984855841071284801314527801
1522127328744184579254348329648010111279968701392471609617828914657408166490485591444164905
Technical details
Node id
n-31b7a62c39b7ba33 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3cde7219e111af75

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:    7,     2;
   4:   28,    20,     3;
   5:   84,   108,    39,     4;
   6:  209,   418,   264,    64,     5;
   7:  455,  1300,  1248,   520,    95,     6;
   8:  896,  3456,  4650,  2900,   900,   132,    7;
   9: 1632,  8160, 14586, 12580,  5780,  1428,  175,   8;
  10: 2793, 17556, 40128, 45448, 28595, 10374, 2128, 224, 9;
%F Formula
The row polynomials satisfy (2*n^4-13*n^3+47*n^2-88*n+60-(2*n^2-7*n+6)*x^2)*P(n,x) = (2*n^4-3*n^3+17*n^2-8*n+(n^4-2*n^3+5*n^2-4*n)*x-(3*n^2-2*n)*x^2-(n^2-n)*x^3)*P(n-1,x) + ((4*n^3-12*n^2+32*n-24)*x+(n^3-3*n^2+2)*x^2-(6*n-6)*x^3-(n-1)*x^4)*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[((2*n^4 - 3*n^3 + 17*n^2 - 8*n + (n^4 - 2*n^3 + 5*n^2 - 4*n)*x - (3*n^2 - 2*n)*x^2 - (n^2 - n)*x^3)*P[n - 1, x] + ((4*n^3 - 12*n^2 + 32*n - 24)*x + (n^3 - 3*n^2 + 2)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 13*n^3 + 47*n^2 - 88*n + 60 - (2*n^2 - 7*n + 6)*x^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] = ((2*n^4 - 3*n^3 + 17*n^2 - 8*n + (n^4 - 2*n^3 + 5*n^2 - 4*n)*x - (3*n^2 - 2*n)*x^2 - (n^2 - n)*x^3)*v[i - 1] + ((4*n^3 - 12*n^2 + 32*n - 24)*x + (n^3 - 3*n^2 + 2)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 13*n^3 + 47*n^2 - 88*n + 60 - (2*n^2 - 7*n + 6)*x^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

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