Real-rooted atlas

A099040 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(4\,x - n - 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-6 - 24\,x - 24\,x^{2} + 8\,n + 16\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 18\,x^{2} - 8\,x^{3} + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
242428
304896144
4024288320632
50028812809602528
600961920480026889504
70001280960016128716834176
80003209600403205017618432118848
9000048005376015052814745646080402624
100000960403202508805160964147201126401335616
1100000161282508801032192165888011264002703364354816
12000002688150528129024038707205068800297369663897613995648
13000000501761032192580608013516800148684807667712149094444432384
1400000071685160965806080236544004460544042172416193822723440640139584512
Technical details
Node id
n-6f9dd8be3d8942d5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e7ae0a9fb7a7fbdd

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: 4;
  2: 4, 24;
  3: 0, 48,  96;
  4: 0, 24, 288,  320;
  5: 0,  0, 288, 1280,  960;
  6: 0,  0,  96, 1920, 4800,  2688;
  7: 0,  0,   0, 1280, 9600, 16128,   7168;
  8: 0,  0,   0,  320, 9600, 40320,  50176,  18432;
  9: 0,  0,   0,    0, 4800, 53760, 150528, 147456, 46080;
%F Formula
The row polynomials satisfy (n^2-n-(4*n-4)*x)*P(n,x) = -(2*n^2-8*n+6-(16*n-24)*x+24*x^2)*P(n-1,x) + ((4*n-4)*x+(2*n-18)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 - (16*n - 24)*x + 24*x^2)*P[n - 1, x] + ((4*n - 4)*x + (2*n - 18)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n^2 - n - (4*n - 4)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 - (16*n - 24)*x + 24*x^2)*v[i - 1] + ((4*n - 4)*x + (2*n - 18)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n^2 - n - (4*n - 4)*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. [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.