Real-rooted atlas

A169803 → D → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
222
3347
441216
5524938
66403682
77609016173
888418080352
9911231524025701
10101445045601501368
11111807561120525362628
12122201080201614002524980
13132641485336031501008499329
1414312198052806300302439217302
Technical details
Node id
n-2da8bd4e7bbc0ddc · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5bc743e1d3ec7e23

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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: 1;
  2: 2;
  3: 3,   4;
  4: 4,  12;
  5: 5,  24,   9;
  6: 6,  40,  36;
  7: 7,  60,  90,  16;
  8: 8,  84, 180,  80;
  9: 9, 112, 315, 240, 25;
%F Formula
The row polynomials satisfy (n^2-1+(n^3-2*n^2+n)*x)*P(n,x) = (n^2+n+(n^3+3*n^2-4*n-8)*x+(2*n^3-2*n^2-4*n)*x^2)*P(n-1,x) - ((n+1)*x+(n^2+5*n+4)*x^2+(4*n^2+4*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (n^3 + 3*n^2 - 4*n - 8)*x + (2*n^3 - 2*n^2 - 4*n)*x^2)*P[n - 1, x] - ((n + 1)*x + (n^2 + 5*n + 4)*x^2 + (4*n^2 + 4*n)*x^3)*D[P[n - 1, x], x])/(n^2 - 1 + (n^3 - 2*n^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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n + (n^3 + 3*n^2 - 4*n - 8)*x + (2*n^3 - 2*n^2 - 4*n)*x^2)*v[i - 1] - ((n + 1)*x + (n^2 + 5*n + 4)*x^2 + (4*n^2 + 4*n)*x^3)*deriv(v[i - 1], x))/(n^2 - 1 + (n^3 - 2*n^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.