Real-rooted atlas

A269939 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(8\,x + 12\,x^{2} - 8\,n x - 16\,n x^{2} + 2\,n^{2} x + 7\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 8\,x^{2} - 6\,x^{3} + n x + 6\,n x^{2} + 5\,n x^{3} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 6\,x - n - 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} + n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\,x\)
\(P_{3}(x) = 2\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2022
302911
402306092
502754205251022
60216819606300567014100
70235776724725010395072765231996
802738273002849001143450189189010810804429360

All 15 rows

Technical details
Node id
n-dac23666be2bdcf0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-146205b457884abc

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 2;
   3: 0, 2,    9;
   4: 0, 2,   30,     60;
   5: 0, 2,   75,    420,     525;
   6: 0, 2,  168,   1960,    6300,     5670;
   7: 0, 2,  357,   7672,   47250,   103950,     72765;
   8: 0, 2,  738,  27300,  284900,  1143450,   1891890,   1081080;
   9: 0, 2, 1503,  91740, 1514975,  9819810,  28693665,  37837800,  18243225;
  10: 0, 2, 3036, 297264, 7436000, 72732660, 332552220, 756756000, 827026200, 344594250;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = ((2*n^2-8*n+8)*x-(n^3-7*n^2+16*n-12)*x^2)*P(n-2,x) + ((n-2)*x-(n^2-6*n+8)*x^2-(n^2-5*n+6)*x^3)*d/dx P(n-2,x) - (n-2-(2*n^2-8*n+6)*x)*P(n-1,x) + ((n-3)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 2*x + 9*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2*x;
P[3, x_] := 2*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 - 8*n + 8)*x - (n^3 - 7*n^2 + 16*n - 12)*x^2)*P[n - 2, x] + ((n - 2)*x - (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] - (n - 2 - (2*n^2 - 8*n + 6)*x)*P[n - 1, x] + ((n - 3)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 2*x + 9*x^2; for(n = 4, N, my(i = n); v[i] = (((2*n^2 - 8*n + 8)*x - (n^3 - 7*n^2 + 16*n - 12)*x^2)*v[i - 2] + ((n - 2)*x - (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) - (n - 2 - (2*n^2 - 8*n + 6)*x)*v[i - 1] + ((n - 3)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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.