Real-rooted atlas

A110441 → basis A007318 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
34424371
4208180484440
590810884568052537
6377658083360920120613990
715200286082124080001620168774843
859776133216121152584801624026042248391700
923099259520064195237945613482029568392028892016205
108805122576352321830422568009811202748484972856163601010243650
113319872108760321545384012559360647584022176005117287872077404401151501183
1212405504449911687169280066328960396403201609171245346568884801188001034052812256703280
1346010816183023616323350272335787520228520800107500032358888328578944145926017248013464624131270306673
141695723527341337601424550912164162585612545369606720416642599027207355462415250752229064024252817160728146247720670
Technical details
Node id
n-b43cd3a6ff6346f0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3ba533e33ff97a8d

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:      1;
  2:      8,      2;
  3:     44,     24,      3;
  4:    208,    180,     48,      4;
  5:    908,   1088,    456,     80,      5;
  6:   3776,   5808,   3360,    920,    120,     6;
  7:  15200,  28608,  21240,   8000,   1620,   168,    7;
  8:  59776, 133216, 121152,  58480,  16240,  2604,  224,   8;
  9: 230992, 595200, 641952, 379456, 134820, 29568, 3920, 288, 9;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (4*n+8+(n+2)*x)*P(n-1,x) + (8+8*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8 + (n + 2)*x)*P[n - 1, x] + (8 + 8*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8 + (n + 2)*x)*v[i - 1] + (8 + 8*x + x^2)*deriv(v[i - 1], x))/(2*n - 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

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