Real-rooted atlas

A052553 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 18\,x - 4\,n + 21\,n x + n^{2} - 8\,n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(2 + 4\,x - 16\,x^{2} - 3\,n - 4\,n x + 28\,n x^{2} + n^{2} - 3\,n^{2} x - 14\,n^{2} x^{2} + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} - 8\,x^{3} - n x - n x^{2} + 12\,n x^{3} - n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
20
311
422
5347
641216
7524938
86403682
97609016173
1088418080352
11911231524025701
12101445045601501368
13111807561120525362628
14122201080201614002524980
15132641485336031501008499329
Technical details
Node id
n-dd98275404c59d4b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5bc743e1d3ec7e23

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -3)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 0;
   2: 0;
   3: 1;
   4: 2;
   5: 3,  4;
   6: 4, 12;
   7: 5, 24,   9;
   8: 6, 40,  36;
   9: 7, 60,  90, 16;
  10: 8, 84, 180, 80;
%F Formula
The row polynomials satisfy (n^2-4*n+3+(n^3-8*n^2+21*n-18)*x)*P(n,x) = (n^2-3*n+2+(n^3-3*n^2-4*n+4)*x+(2*n^3-14*n^2+28*n-16)*x^2)*P(n-1,x) - ((n-1)*x+(n^2+n-2)*x^2+(4*n^2-12*n+8)*x^3)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 3*n + 2 + (n^3 - 3*n^2 - 4*n + 4)*x + (2*n^3 - 14*n^2 + 28*n - 16)*x^2)*P[n - 1, x] - ((n - 1)*x + (n^2 + n - 2)*x^2 + (4*n^2 - 12*n + 8)*x^3)*D[P[n - 1, x], x])/(n^2 - 4*n + 3 + (n^3 - 8*n^2 + 21*n - 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = ((n^2 - 3*n + 2 + (n^3 - 3*n^2 - 4*n + 4)*x + (2*n^3 - 14*n^2 + 28*n - 16)*x^2)*v[i - 1] - ((n - 1)*x + (n^2 + n - 2)*x^2 + (4*n^2 - 12*n + 8)*x^3)*deriv(v[i - 1], x))/(n^2 - 4*n + 3 + (n^3 - 8*n^2 + 21*n - 18)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.