Real-rooted atlas

A054142 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + x^{2} + 5\,n - 4\,n x - 2\,n x^{2} - 3\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-2 + 2\,x + x^{2} + 3\,n - 6\,n x - 5\,n x^{2} - n x^{3} + n^{2} + n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x - 18\,x^{2} - 8\,x^{3} - x^{4} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
245110
37168132
411403611199
5168511964141299
622161322264100171887
7292807568704951442012595

All 15 rows

Technical details
Node id
n-310c5e0cb618ceb4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f1305d7953ba0815

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:  1;
  1:  2,   1;
  2:  4,   5,    1;
  3:  7,  16,    8,    1;
  4: 11,  40,   36,   11,    1;
  5: 16,  85,  119,   64,   14,    1;
  6: 22, 161,  322,  264,  100,   17,    1;
  7: 29, 280,  756,  870,  495,  144,   20,   1;
  8: 37, 456, 1596, 2442, 1925,  832,  196,  23,  1;
  9: 46, 705, 3102, 6072, 6292, 3731, 1295, 256, 26, 1;
%F Formula
The row polynomials satisfy (2*n^3-3*n^2+5*n-2-(4*n-2)*x-(2*n-1)*x^2)*P(n,x) = (2*n^3+n^2+3*n-2+(n^3+n^2-6*n+2)*x-(5*n-1)*x^2-n*x^3)*P(n-1,x) + ((4*n^2-8)*x+(n^2-18)*x^2-8*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + n^2 + 3*n - 2 + (n^3 + n^2 - 6*n + 2)*x - (5*n - 1)*x^2 - n*x^3)*P[n - 1, x] + ((4*n^2 - 8)*x + (n^2 - 18)*x^2 - 8*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 5*n - 2 - (4*n - 2)*x - (2*n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + n^2 + 3*n - 2 + (n^3 + n^2 - 6*n + 2)*x - (5*n - 1)*x^2 - n*x^3)*v[i - 1] + ((4*n^2 - 8)*x + (n^2 - 18)*x^2 - 8*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 5*n - 2 - (4*n - 2)*x - (2*n - 1)*x^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. [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