Real-rooted atlas

A008459 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(2 + 8\,x - 6\,n - 24\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} - 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2243256
3120384192696
45602880384010248304
52520179204032030720512096600
611088100800322560430080215040245761104144
748048532224221760043008003870720137625611468812460336

All 15 rows

Technical details
Node id
n-5c524fa039d681e9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-353d76cb8661f697

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:     24,       32;
  3:    120,      384,      192;
  4:    560,     2880,     3840,      1024;
  5:   2520,    17920,    40320,     30720,      5120;
  6:  11088,   100800,   322560,    430080,    215040,     24576;
  7:  48048,   532224,  2217600,   4300800,   3870720,   1376256,    114688;
  8: 205920,  2690688, 13837824,  35481600,  47308800,  30965760,   8257536,   524288;
  9: 875160, 13178880, 80720640, 258306048, 461260800, 454164480, 227082240, 47185920, 2359296;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (4*n^2-6*n+2+(12*n^2-24*n+8)*x)*P(n-1,x) - ((2*n-1)*x+(8*n-4)*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[((4*n^2 - 6*n + 2 + (12*n^2 - 24*n + 8)*x)*P[n - 1, x] - ((2*n - 1)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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 - 6*n + 2 + (12*n^2 - 24*n + 8)*x)*v[i - 1] - ((2*n - 1)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023