Real-rooted atlas

A128099 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,x - 2\,n + 16\,n x + n^{2} - 10\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(-4\,x - n - 14\,n x + 24\,n x^{2} + n^{2} - 32\,n^{2} x^{2} + 2\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-n x - 6\,n x^{2} + 16\,n x^{3} - 2\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
222
344
461622
584856
6109672178
712160288460
8142407202561230
916336144012803072
1018448252038408007626
112057640328960480018388
122272060481792016800230443814
13248808640322564480016128102728
142610561188053760100800645126272238306
Technical details
Node id
n-d1a6b780d2838447 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2fe8080482827c57

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  0;
  2:  2;
  3:  4;
  4:  6,  16;
  5:  8,  48;
  6: 10,  96,   72;
  7: 12, 160,  288;
  8: 14, 240,  720,  256;
  9: 16, 336, 1440, 1280;
%F Formula
The row polynomials satisfy (n^2-2*n+(2*n^3-10*n^2+16*n-8)*x)*P(n,x) = (n^2-n+(2*n^3-14*n-4)*x+(8*n^3-32*n^2+24*n)*x^2)*P(n-1,x) - (n*x+(2*n^2+6*n)*x^2+(16*n^2-16*n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + (2*n^3 - 14*n - 4)*x + (8*n^3 - 32*n^2 + 24*n)*x^2)*P[n - 1, x] - (n*x + (2*n^2 + 6*n)*x^2 + (16*n^2 - 16*n)*x^3)*D[P[n - 1, x], x])/(n^2 - 2*n + (2*n^3 - 10*n^2 + 16*n - 8)*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] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 - n + (2*n^3 - 14*n - 4)*x + (8*n^3 - 32*n^2 + 24*n)*x^2)*v[i - 1] - (n*x + (2*n^2 + 6*n)*x^2 + (16*n^2 - 16*n)*x^3)*deriv(v[i - 1], x))/(n^2 - 2*n + (2*n^3 - 10*n^2 + 16*n - 8)*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

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.