Real-rooted atlas

A112493 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
34610
4115045106
5262606304201336
657109252508820472519944
712040743410410710013860062370346368
824714142191415990220230422524324309459456878624

All 15 rows

Technical details
Node id
n-ec648b55f50a233f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b73aa855d7a17944

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 -2)
  • 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:    1;
   3:    4,      6;
   4:   11,     50,      45;
   5:   26,    260,     630,      420;
   6:   57,   1092,    5250,     8820,      4725;
   7:  120,   4074,   34104,   107100,    138600,     62370;
   8:  247,  14142,  191415,   990220,   2304225,   2432430,     945945;
   9:  502,  46872,  977790,  7758520,  28678650,  52972920,   47297250,   16216200;
  10: 1013, 150656, 4684548, 54437240, 298147850, 852611760, 1305404100, 1010809800, 310134825;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(2*n-2+(2*n^2-4*n+2)*x+(n^3-4*n^2+5*n-2)*x^2)*P(n-2,x) - ((n-1)*x+(n^2-2*n+1)*x^2+(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (3*n-5+(2*n^2-4*n)*x)*P(n-1,x) + ((n-2)*x+(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (2*n^2 - 4*n + 2)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*P[n - 2, x] - ((n - 1)*x + (n^2 - 2*n + 1)*x^2 + (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (3*n - 5 + (2*n^2 - 4*n)*x)*P[n - 1, x] + ((n - 2)*x + (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n - 2 + (2*n^2 - 4*n + 2)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*v[i - 2] - ((n - 1)*x + (n^2 - 2*n + 1)*x^2 + (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (3*n - 5 + (2*n^2 - 4*n)*x)*v[i - 1] + ((n - 2)*x + (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.