Real-rooted atlas

A099040 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n x - 4\,n x^{2} + 2\,n x^{3} - n^{2} + 2\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12\,x - 16\,x^{2} - 4\,x^{3} + 8\,x^{4} - 6\,n - 4\,n x + 4\,n x^{2} + 4\,n x^{3} + 8\,n^{2} + 8\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} - 4\,x^{3} + 8\,x^{4} + 4\,x^{5} - 4\,n x - 2\,n x^{2} + 8\,n x^{3} + 4\,n x^{4} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
24281244
304812032200
402430041680820
500288137612801923136
6009619525280364844811424
700012809920180489856102440128

All 15 rows

Technical details
Node id
n-73ca25a1e5b58e01 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0ebd3d19d515d969

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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: 1;
  1: 4,  4;
  2: 4, 28,  12;
  3: 0, 48, 120,   32;
  4: 0, 24, 300,  416,   80;
  5: 0,  0, 288, 1376, 1280,   192;
  6: 0,  0,  96, 1952, 5280,  3648,    448;
  7: 0,  0,   0, 1280, 9920, 18048,   9856,   1024;
  8: 0,  0,   0,  320, 9680, 42240,  56896,  25600,  2304;
  9: 0,  0,   0,    0, 4800, 54720, 159488, 168960, 64512, 5120;
%F Formula
The row polynomials satisfy (n^3-n^2-2*n*x+(2*n^2-4*n)*x^2+2*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(8*n^2-4*n-12)*x+(4*n^2-4*n+16)*x^2-(4*n-4)*x^3-8*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(2*n^2-2*n-8)*x^2+(8*n-4)*x^3+(4*n+8)*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 8*n^2 + 6*n - (8*n^2 - 4*n - 12)*x + (4*n^2 - 4*n + 16)*x^2 - (4*n - 4)*x^3 - 8*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (2*n^2 - 2*n - 8)*x^2 + (8*n - 4)*x^3 + (4*n + 8)*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 - 2*n*x + (2*n^2 - 4*n)*x^2 + 2*n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 - 8*n^2 + 6*n - (8*n^2 - 4*n - 12)*x + (4*n^2 - 4*n + 16)*x^2 - (4*n - 4)*x^3 - 8*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (2*n^2 - 2*n - 8)*x^2 + (8*n - 4)*x^3 + (4*n + 8)*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 - 2*n*x + (2*n^2 - 4*n)*x^2 + 2*n*x^3)); 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. [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.