Real-rooted atlas

A207823 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 8\,n\right)\,P_{n}(x) = \left(40 + 32\,x + 20\,n + 16\,n x\right)\,P_{n-1}(x) + \left(21 + 40\,x + 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2403272
3292480192964
4188047043840102411448
5113123808047232256005120127344
665240276192460800378880153600245761359288
7365484186480039139204326400265728086016011468814102732

All 15 rows

Technical details
Node id
n-6a232ac5154579b3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-81dee71be5720b4c

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:       40,       32;
  3:      292,      480,       192;
  4:     1880,     4704,      3840,      1024;
  5:    11312,    38080,     47232,     25600,      5120;
  6:    65240,   276192,    460800,    378880,    153600,     24576;
  7:   365484,  1864800,   3913920,   4326400,   2657280,    860160,    114688;
  8:  2004600, 11971680,  30297600,  42127360,  34764800,  17031168,   4587520,   524288;
  9: 10819020, 74030400, 219404160, 367897600, 381803520, 251166720, 102301696, 23592960, 2359296;
%F Formula
The row polynomials satisfy (8*n-8)*P(n,x) = (20*n+40+(16*n+32)*x)*P(n-1,x) + (21+40*x+16*x^2)*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[((20*n + 40 + (16*n + 32)*x)*P[n - 1, x] + (21 + 40*x + 16*x^2)*D[P[n - 1, x], x])/(8*n - 8)]];
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] = ((20*n + 40 + (16*n + 32)*x)*v[i - 1] + (21 + 40*x + 16*x^2)*deriv(v[i - 1], x))/(8*n - 8)); 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