Real-rooted atlas

A056241 → D

nonnegative coefficients

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

Recurrence

\(\left(-12\,x + 6\,x^{2} - n + 17\,n x - 10\,n x^{2} + n^{2} - 5\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(n + 15\,n x + 3\,n x^{2} - 22\,n x^{3} + n^{2} - 5\,n^{2} x + n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(2\,n x - 6\,n x^{2} - 6\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2325
3612321
4103830482
51590135605305
62118042336010561095
7283221071142880516873829
8365322352442839201596252813124
9458284662116281453593242898360944289
10551230855027060447654059019950492049510147625
11661760147515742012034014440810048539336792066011487157
12782442242221132562913904427344079462265127266612210858121594326
1391330238181210496649220120985814115011038752473616127270181611092135181553
14105436858149372372135135030155584313589402074424324309309302132132620813651416740395
Technical details
Node id
n-07a074fde62a4797 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6c5bc31b8ab24985

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:  3,   2;
  3:  6,  12,    3;
  4: 10,  38,   30,     4;
  5: 15,  90,  135,    60,     5;
  6: 21, 180,  423,   360,   105,    6;
  7: 28, 322, 1071,  1428,   805,  168,    7;
  8: 36, 532, 2352,  4428,  3920, 1596,  252,   8;
  9: 45, 828, 4662, 11628, 14535, 9324, 2898, 360, 9;
%F Formula
The row polynomials satisfy (n^2-n-(5*n^2-17*n+12)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (n^2+n-(5*n^2-15*n)*x+(n^2+3*n)*x^2+(12*n^2-22*n)*x^3)*P(n-1,x) + (2*n*x-6*n*x^2-6*n*x^3-8*n*x^4)*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[((n^2 + n - (5*n^2 - 15*n)*x + (n^2 + 3*n)*x^2 + (12*n^2 - 22*n)*x^3)*P[n - 1, x] + (2*n*x - 6*n*x^2 - 6*n*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(n^2 - n - (5*n^2 - 17*n + 12)*x + (4*n^2 - 10*n + 6)*x^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] = ((n^2 + n - (5*n^2 - 15*n)*x + (n^2 + 3*n)*x^2 + (12*n^2 - 22*n)*x^3)*v[i - 1] + (2*n*x - 6*n*x^2 - 6*n*x^3 - 8*n*x^4)*deriv(v[i - 1], x))/(n^2 - n - (5*n^2 - 17*n + 12)*x + (4*n^2 - 10*n + 6)*x^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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.