Real-rooted atlas

A060920 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - 10\,x + 5\,n + 35\,n x - n^{2} - 35\,n^{2} x - 2\,n^{3} + 10\,n^{3} x\right)\,P_{n}(x) = \left(-8\,x - 10\,x^{2} + 3\,n + 16\,n x - 25\,n x^{2} - 50\,n x^{3} - 5\,n^{2} - 43\,n^{2} x + 15\,n^{2} x^{2} + 75\,n^{2} x^{3} - 2\,n^{3} + 11\,n^{3} x - 25\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 9\,x^{2} - 5\,x^{3} - 20\,n x^{2} - 45\,n x^{3} - 25\,n x^{4} - 4\,n^{2} x + 11\,n^{2} x^{2} + 40\,n^{2} x^{3} + 25\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
251015
39403988
414102213136465
5202106999404452314
627380178237683720139811075
7356303885114401761013710427051580

All 15 rows

Technical details
Node id
n-caba3b00effe5aa9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1e104af877c3f00c

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:  2;
  2:  5,   10;
  3:  9,   40,    39;
  4: 14,  102,   213,   136;
  5: 20,  210,   699,   940,    445;
  6: 27,  380,  1782,  3768,   3720,   1398;
  7: 35,  630,  3885, 11440,  17610,  13710,   4270;
  8: 44,  980,  7602, 29140,  62010,  74838,  48055,  12776;
  9: 54, 1452, 13734, 65628, 180610, 299778, 297129, 162272, 37629;
%F Formula
The row polynomials satisfy -(2*n^3+n^2-5*n+2-(10*n^3-35*n^2+35*n-10)*x)*P(n,x) = -(2*n^3+5*n^2-3*n-(11*n^3-43*n^2+16*n-8)*x-(15*n^2-25*n-10)*x^2+(25*n^3-75*n^2+50*n)*x^3)*P(n-1,x) - ((4*n^2+4)*x-(11*n^2-20*n-9)*x^2-(40*n^2-45*n-5)*x^3-(25*n^2-25*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 5*n^2 - 3*n - (11*n^3 - 43*n^2 + 16*n - 8)*x - (15*n^2 - 25*n - 10)*x^2 + (25*n^3 - 75*n^2 + 50*n)*x^3)*P[n - 1, x] - ((4*n^2 + 4)*x - (11*n^2 - 20*n - 9)*x^2 - (40*n^2 - 45*n - 5)*x^3 - (25*n^2 - 25*n)*x^4)*D[P[n - 1, x], x])/(-2*n^3 - n^2 + 5*n - 2 + (10*n^3 - 35*n^2 + 35*n - 10)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 + 5*n^2 - 3*n - (11*n^3 - 43*n^2 + 16*n - 8)*x - (15*n^2 - 25*n - 10)*x^2 + (25*n^3 - 75*n^2 + 50*n)*x^3)*v[i - 1] - ((4*n^2 + 4)*x - (11*n^2 - 20*n - 9)*x^2 - (40*n^2 - 45*n - 5)*x^3 - (25*n^2 - 25*n)*x^4)*deriv(v[i - 1], x))/(-2*n^3 - n^2 + 5*n - 2 + (10*n^3 - 35*n^2 + 35*n - 10)*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