Real-rooted atlas

A391403 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 3\,n x + 4\,n x^{2} + n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(12 + 8\,x - 20\,x^{2} - 16\,x^{3} - 22\,n - 31\,n x + 8\,n x^{2} + 16\,n x^{3} + 3\,n^{2} + 20\,n^{2} x + 16\,n^{2} x^{2} + 5\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-6 - 44\,x - 102\,x^{2} - 96\,x^{3} - 32\,x^{4} - 18\,n - 66\,n x - 80\,n x^{2} - 32\,n x^{3} - 6\,n^{2} - 14\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2527
313619
44336685
513114030301
6411570210201211
71275214210501404607

All 15 rows

Technical details
Node id
n-389e361771f4b4ac · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bfa3eeae29a25dff

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:     1;
  2:     5,     2;
  3:    13,     6;
  4:    43,    36,     6;
  5:   131,   140,    30;
  6:   411,   570,   210,   20;
  7:  1275,  2142,  1050,  140;
  8:  3963,  7896,  5040, 1120,  70;
  9: 12283, 28296, 22176, 6720, 630;
%F Formula
The row polynomials satisfy (n^3+n^2-4*n+(4*n^2-3*n)*x+4*n*x^2)*P(n,x) = (5*n^3+3*n^2-22*n+12+(4*n^3+20*n^2-31*n+8)*x+(16*n^2+8*n-20)*x^2+(16*n-16)*x^3)*P(n-1,x) - (6*n^2+18*n+6+(14*n^2+66*n+44)*x+(8*n^2+80*n+102)*x^2+(32*n+96)*x^3+32*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[((5*n^3 + 3*n^2 - 22*n + 12 + (4*n^3 + 20*n^2 - 31*n + 8)*x + (16*n^2 + 8*n - 20)*x^2 + (16*n - 16)*x^3)*P[n - 1, x] - (6*n^2 + 18*n + 6 + (14*n^2 + 66*n + 44)*x + (8*n^2 + 80*n + 102)*x^2 + (32*n + 96)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 - 4*n + (4*n^2 - 3*n)*x + 4*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] = ((5*n^3 + 3*n^2 - 22*n + 12 + (4*n^3 + 20*n^2 - 31*n + 8)*x + (16*n^2 + 8*n - 20)*x^2 + (16*n - 16)*x^3)*v[i - 1] - (6*n^2 + 18*n + 6 + (14*n^2 + 66*n + 44)*x + (8*n^2 + 80*n + 102)*x^2 + (32*n + 96)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 - 4*n + (4*n^2 - 3*n)*x + 4*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

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023