Real-rooted atlas

A085478 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(20 - 18\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12 + 46\,x - 14\,n - 50\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x + 12\,x^{2} - 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3325
4512320
573030471
6956105605235
711902522801056744
81313249584063016872285

All 15 rows

Technical details
Node id
n-cd6dc20f7cfa6d5c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5848b624a80b5ea0

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:  3,   2;
   4:  5,  12,    3;
   5:  7,  30,   30,    4;
   6:  9,  56,  105,   60,    5;
   7: 11,  90,  252,  280,  105,    6;
   8: 13, 132,  495,  840,  630,  168,    7;
   9: 15, 182,  858, 1980, 2310, 1260,  252,   8;
  10: 17, 240, 1365, 4004, 6435, 5544, 2310, 360, 9;
%F Formula
The row polynomials satisfy (4*n^2-18*n+20)*P(n,x) = (4*n^2-14*n+12+(12*n^2-50*n+46)*x)*P(n-1,x) - ((2*n-3)*x+(8*n-12)*x^2)*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[((4*n^2 - 14*n + 12 + (12*n^2 - 50*n + 46)*x)*P[n - 1, x] - ((2*n - 3)*x + (8*n - 12)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 18*n + 20)]];
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] = ((4*n^2 - 14*n + 12 + (12*n^2 - 50*n + 46)*x)*v[i - 1] - ((2*n - 3)*x + (8*n - 12)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 18*n + 20)); 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

Same fingerprint ?

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