Real-rooted atlas

A345013 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 + n^{2}\right)\,P_{n}(x) = \left(2\,n + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-x - x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2201232
310512030255
4504840420601824
5231050403780112010512355
610296277202772012600252016881024
745045144144180180110880346505040252520191

All 15 rows

Technical details
Node id
n-fabe4ac480a936a7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5cd009bfc7bc0bb1

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:      3;
  2:     20,      12;
  3:    105,     120,      30;
  4:    504,     840,     420,      60;
  5:   2310,    5040,    3780,    1120,     105;
  6:  10296,   27720,   27720,   12600,    2520,     168;
  7:  45045,  144144,  180180,  110880,   34650,    5040,    252;
  8: 194480,  720720, 1081080,  840840,  360360,   83160,   9240,   360;
  9: 831402, 3500640, 6126120, 5765760, 3153150, 1009008, 180180, 15840, 495;
%F Formula
The row polynomials satisfy (n^2-1)*P(n,x) = (4*n^2+2*n+3*n^2*x)*P(n-1,x) - ((2*n+1)*x+(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + 3*n^2*x)*P[n - 1, x] - ((2*n + 1)*x + (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n + 3*n^2*x)*v[i - 1] - ((2*n + 1)*x + (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.