Real-rooted atlas

A111125 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(18 - 21\,n + 6\,n^{2}\right)\,P_{n}(x) = \left(-6\,n - 9\,n x + 9\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x) + \left(-5 - 18\,x - 9\,x^{2} + 5\,n + 18\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
233
3211839
49318081354
5345108010533242802
61170513079385184121520637
73762213304584646980230854374145377
811679814322255853191402442159622815309993588
935367293040997353181278019063351163484382725524886643572
1010513510098904085154910861212272715102570305205060146966417714743690407
1130808533674401580261441853672690691057378938051132060222024245491557590490283606827
128925481094094058476168179662860351885870459759888406453950240447528912307052007666019486171821775734
132561796348106682088601207310395801660324230257227941627773893982092696560107971096536374184072098829637729211601890694
1472961231088618407247521982848896360737048524513234245588169286922001556012008810210753080467018541014146959422550916802072619973354801953
1520644185335551590245527475410714658292311344928556366420423094328546760102628894104820679535724768265799019583600850538881174089122655766961566460963479045
Technical details
Node id
n-330a3ebfbeeab417 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-06e11ce7c0cd5072

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:      3;
   3:     21,      18;
   4:     93,     180,      81;
   5:    345,    1080,    1053,     324;
   6:   1170,    5130,    7938,    5184,     1215;
   7:   3762,   21330,   45846,   46980,    23085,     4374;
   8:  11679,   81432,  225585,  319140,   244215,    96228,   15309;
   9:  35367,  293040,  997353, 1812780,  1906335,  1163484,  382725,   52488;
  10: 105135, 1009890, 4085154, 9108612, 12272715, 10257030, 5205060, 1469664, 177147;
%F Formula
The row polynomials satisfy (6*n^2-21*n+18)*P(n,x) = (9*n^2-6*n+(9*n^2-9*n)*x)*P(n-1,x) + (5*n-5+(18*n-18)*x+(9*n-9)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 6*n + (9*n^2 - 9*n)*x)*P[n - 1, x] + (5*n - 5 + (18*n - 18)*x + (9*n - 9)*x^2)*D[P[n - 1, x], x])/(6*n^2 - 21*n + 18)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((9*n^2 - 6*n + (9*n^2 - 9*n)*x)*v[i - 1] + (5*n - 5 + (18*n - 18)*x + (9*n - 9)*x^2)*deriv(v[i - 1], x))/(6*n^2 - 21*n + 18)); 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

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