Real-rooted atlas

A113953 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(x - n - n x + n^{2}\right)\,P_{n}(x) = \left(-12 - 12\,x - 3\,x^{2} + 16\,n + 8\,n x - 4\,n^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 9\,x^{2} - x^{3} + 8\,n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
24610
30241236
40247220116
50014416030334
6009648030042918
70006401200504562400
800032024002520784726096
9000024006720470411529015066
10000096010080156808064162011036514
11000008064313603225612960220013286972
12000002688376328064060480198002904156204300
1300000025088129024181440105600290403744182474118
1400000071681290243628803696001742404118447322101089038
Technical details
Node id
n-26bcb093c0c8d51e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9240d5b41e77e9e3

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: 4,  6;
  3: 0, 24,  12;
  4: 0, 24,  72,  20;
  5: 0,  0, 144, 160,   30;
  6: 0,  0,  96, 480,  300,   42;
  7: 0,  0,   0, 640, 1200,  504,   56;
  8: 0,  0,   0, 320, 2400, 2520,  784,   72;
  9: 0,  0,   0,   0, 2400, 6720, 4704, 1152, 90;
%F Formula
The row polynomials satisfy (n^2-n-(n-1)*x)*P(n,x) = -(4*n^2-16*n+12-(8*n-12)*x+3*x^2)*P(n-1,x) + ((8*n-8)*x+(n-9)*x^2-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[(-(4*n^2 - 16*n + 12 - (8*n - 12)*x + 3*x^2)*P[n - 1, x] + ((8*n - 8)*x + (n - 9)*x^2 - x^3)*D[P[n - 1, x], x])/(n^2 - n - (n - 1)*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] = (-(4*n^2 - 16*n + 12 - (8*n - 12)*x + 3*x^2)*v[i - 1] + ((8*n - 8)*x + (n - 9)*x^2 - x^3)*deriv(v[i - 1], x))/(n^2 - n - (n - 1)*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.