Real-rooted atlas

A208509 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 - 6\,x + 8\,n + n x^{2} + 2\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(4 - 34\,x - 2\,x^{3} + 8\,n + 16\,n x + 13\,n x^{2} - n x^{3} - 2\,n^{2} x + 2\,n^{2} x^{2} + 2\,n^{2} x^{3} - 4\,n^{3} x - 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 20\,x^{2} + 8\,x^{3} - 4\,x^{4} - 16\,n x^{2} + 22\,n x^{3} - 6\,n x^{4} - 8\,n^{2} x^{2} + 10\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
2527
371017
4928340
511602192
613110814208
71518223136464
81728054617651024
9194081134624552240
10215702142180032564864
11237703762448813757810496
122510126237100324675540722528
1327130098672059213585265210548128
14291638150153946835035103748338102400
Technical details
Node id
n-95642cbad4635d0e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bd0fa241afe43c8f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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:  1;
  1:  3;
  2:  5,   2;
  3:  7,  10;
  4:  9,  28,    3;
  5: 11,  60,   21;
  6: 13, 110,   81,   4;
  7: 15, 182,  231,  36;
  8: 17, 280,  546, 176,  5;
  9: 19, 408, 1134, 624, 55;
%F Formula
The row polynomials satisfy (8*n-4-(4*n^3-2*n^2+6)*x+(n^3+2*n^2+n)*x^2)*P(n,x) = (8*n+4-(4*n^3+2*n^2-16*n+34)*x-(3*n^3-2*n^2-13*n)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) + (16*x-(8*n^2+16*n+20)*x^2+(10*n^2+22*n+8)*x^3-(2*n^2+6*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 4 - (4*n^3 + 2*n^2 - 16*n + 34)*x - (3*n^3 - 2*n^2 - 13*n)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] + (16*x - (8*n^2 + 16*n + 20)*x^2 + (10*n^2 + 22*n + 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(8*n - 4 - (4*n^3 - 2*n^2 + 6)*x + (n^3 + 2*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 4 - (4*n^3 + 2*n^2 - 16*n + 34)*x - (3*n^3 - 2*n^2 - 13*n)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] + (16*x - (8*n^2 + 16*n + 20)*x^2 + (10*n^2 + 22*n + 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(8*n - 4 - (4*n^3 - 2*n^2 + 6)*x + (n^3 + 2*n^2 + n)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.