Real-rooted atlas

reversed rows → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
233
3448
451015
5618630
67282156
7840488104
89549036189
9107015010010340
10118823122055605
1112108336420180121068
1213130468728455781872
13141546301176980294143262
1415180825180018908401055655
Technical details
Node id
n-3a607c1aa171bfc2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8de4b75c4559d58b

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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:  2;
  2:  3;
  3:  4,  4;
  4:  5, 10;
  5:  6, 18,   6;
  6:  7, 28,  21;
  7:  8, 40,  48,   8;
  8:  9, 54,  90,  36;
  9: 10, 70, 150, 100, 10;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (n^2-1+(2*n^2-2*n-4)*x)*P(n-1,x) - ((n+1)*x+(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 1 + (2*n^2 - 2*n - 4)*x)*P[n - 1, x] - ((n + 1)*x + (4*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 1 + (2*n^2 - 2*n - 4)*x)*v[i - 1] - ((n + 1)*x + (4*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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. [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.