Real-rooted atlas

A116414 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x - 6\,n - 6\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(-3\,x + 8\,x^{2} + 8\,x^{3} - 2\,n - 31\,n x - 44\,n x^{2} - 12\,n x^{3} + 4\,n^{2} + 26\,n^{2} x + 20\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 8\,x^{3} + 4\,x^{4} - 2\,n x - 17\,n x^{2} - 16\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
282634
31284120216
4161745524841226
5202961488292418206548
6244503120981613680655833648
728636564024680553205906422960168328

All 15 rows

Technical details
Node id
n-9b6770198ef21429 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a450f888131ea287

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,   26;
  3: 12,   84,   120;
  4: 16,  174,   552,   484;
  5: 20,  296,  1488,  2924,   1820;
  6: 24,  450,  3120,  9816,  13680,   6558;
  7: 28,  636,  5640, 24680,  55320,  59064,   22960;
  8: 32,  854,  9240, 52060, 163120, 280428,  241136,  78728;
  9: 36, 1104, 14112, 97524, 395220, 952344, 1317456, 944808, 265716;
%F Formula
The row polynomials satisfy (4*n^2-6*n+2+(2*n^2-6*n+4)*x)*P(n,x) = (4*n^2-2*n+(26*n^2-31*n-3)*x+(20*n^2-44*n+8)*x^2+(4*n^2-12*n+8)*x^3)*P(n-1,x) - (2*n*x+(17*n-1)*x^2+(16*n-8)*x^3+(4*n-4)*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 + (26*n^2 - 31*n - 3)*x + (20*n^2 - 44*n + 8)*x^2 + (4*n^2 - 12*n + 8)*x^3)*P[n - 1, x] - (2*n*x + (17*n - 1)*x^2 + (16*n - 8)*x^3 + (4*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (2*n^2 - 6*n + 4)*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 + (26*n^2 - 31*n - 3)*x + (20*n^2 - 44*n + 8)*x^2 + (4*n^2 - 12*n + 8)*x^3)*v[i - 1] - (2*n*x + (17*n - 1)*x^2 + (16*n - 8)*x^3 + (4*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (2*n^2 - 6*n + 4)*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. [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