Real-rooted atlas

A113214 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 2\,n x - 3\,n^{2} - 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(16 + 4\,x - 28\,n - 24\,n x - 6\,n x^{2} + 14\,n^{2} + 8\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2} - 12\,n x - 11\,n x^{2} - 2\,n x^{3} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
26612
34241240
40306020110
501210812030270
6008428021042616
70024320600336561336

All 15 rows

Technical details
Node id
n-64e45044a07aabed · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5982380e7440ea27

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: 2;
  2: 6,  6;
  3: 4, 24,  12;
  4: 0, 30,  60,  20;
  5: 0, 12, 108, 120,  30;
  6: 0,  0,  84, 280, 210,   42;
  7: 0,  0,  24, 320, 600,  336,   56;
  8: 0,  0,   0, 180, 900, 1134,  504,  72;
  9: 0,  0,   0,  40, 750, 2100, 1960, 720, 90;
%F Formula
The row polynomials satisfy (n^3-3*n^2+2*n-(2*n^2-2*n)*x)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(8*n^2-24*n+4)*x+6*n*x^2)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2-11*n+2)*x^2-2*n*x^3)*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[(-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n^2 - 24*n + 4)*x + 6*n*x^2)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 - 11*n + 2)*x^2 - 2*n*x^3)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n - (2*n^2 - 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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n^2 - 24*n + 4)*x + 6*n*x^2)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 - 11*n + 2)*x^2 - 2*n*x^3)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n - (2*n^2 - 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

Same fingerprint ?

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. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.