Real-rooted atlas

A099093 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(6\,x - n - 6\,n x + n^{2}\right)\,P_{n}(x) = \left(-6 - 36\,x - 54\,x^{2} + 8\,n + 24\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 27\,x^{2} - 18\,x^{3} + 4\,n x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
265460
30108324432
405497216202646
5009726480729014742
6003249720364503061877112
7000648072900183708122472385560
80001620729004592708573044723921863486
90000364506123602571912377913617714708771328
100000729045927042865201322697615943230649539040418676
1100000183708428652026453952637729206495390023383404183034404
12000003061825719123306744014880348029229255025721744482904796816888240
130000008573042645395222320522077944680012860872209948575522901667863601074834
14000000122472132269762232052201364031900385826166054717165363772168218100442349015707156472
Technical details
Node id
n-6f25b129555d52cc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-213b874b48da7e3a

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: 6;
  2: 6,  54;
  3: 0, 108, 324;
  4: 0,  54, 972, 1620;
  5: 0,   0, 972, 6480,  7290;
  6: 0,   0, 324, 9720, 36450,  30618;
  7: 0,   0,   0, 6480, 72900, 183708,  122472;
  8: 0,   0,   0, 1620, 72900, 459270,  857304,  472392;
  9: 0,   0,   0,    0, 36450, 612360, 2571912, 3779136, 1771470;
%F Formula
The row polynomials satisfy (n^2-n-(6*n-6)*x)*P(n,x) = -(2*n^2-8*n+6-(24*n-36)*x+54*x^2)*P(n-1,x) + ((4*n-4)*x+(3*n-27)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 - (24*n - 36)*x + 54*x^2)*P[n - 1, x] + ((4*n - 4)*x + (3*n - 27)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(n^2 - n - (6*n - 6)*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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 - (24*n - 36)*x + 54*x^2)*v[i - 1] + ((4*n - 4)*x + (3*n - 27)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(n^2 - n - (6*n - 6)*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.