Real-rooted atlas

A113214 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(3\,n + n x - 4\,n^{2} - n^{2} x + n^{3}\right)\,P_{n}(x) = \left(20 + 4\,x - 45\,n - 19\,n x - 2\,n x^{2} + 19\,n^{2} + 5\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(10 + 12\,x + 2\,x^{2} - 20\,n - 29\,n x - 10\,n x^{2} - n x^{3} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2527
31314330
42952274112
560150129445388
61183804592566561284
7225886138310844459074120

All 15 rows

Technical details
Node id
n-9c70b79354a3dfea · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4f19a2187fbd0146

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:   5,    2;
  3:  13,   14,    3;
  4:  29,   52,   27,     4;
  5:  60,  150,  129,    44,    5;
  6: 118,  380,  459,   256,   65,    6;
  7: 225,  886, 1383,  1084,  445,   90,    7;
  8: 419, 1952, 3741,  3796, 2185,  708,  119,   8;
  9: 767, 4126, 9381, 11712, 8730, 3954, 1057, 152, 9;
%F Formula
The row polynomials satisfy (n^3-4*n^2+3*n-(n^2-n)*x)*P(n,x) = -(2*n^3-19*n^2+45*n-20-(5*n^2-19*n+4)*x+2*n*x^2)*P(n-1,x) + (5*n^2-20*n+10+(6*n^2-29*n+12)*x+(n^2-10*n+2)*x^2-n*x^3)*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[(-(2*n^3 - 19*n^2 + 45*n - 20 - (5*n^2 - 19*n + 4)*x + 2*n*x^2)*P[n - 1, x] + (5*n^2 - 20*n + 10 + (6*n^2 - 29*n + 12)*x + (n^2 - 10*n + 2)*x^2 - n*x^3)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 3*n - (n^2 - n)*x)]];
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] = (-(2*n^3 - 19*n^2 + 45*n - 20 - (5*n^2 - 19*n + 4)*x + 2*n*x^2)*v[i - 1] + (5*n^2 - 20*n + 10 + (6*n^2 - 29*n + 12)*x + (n^2 - 10*n + 2)*x^2 - n*x^3)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 3*n - (n^2 - n)*x)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.