Real-rooted atlas

A078812 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,n + 8\,n x - 8\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(4 + 4\,x + 4\,n - n x^{2} - 2\,n^{2} x + 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2} - 4\,n x^{2} - n x^{3} + 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3448
41024943
520847216192
63522432416025768
7565041080880300362856
88410082970352019505044910085

All 15 rows

Technical details
Node id
n-42b725eced4ff912 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f3b2dfd067262f26

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,    4;
   4:  10,   24,     9;
   5:  20,   84,    72,    16;
   6:  35,  224,   324,   160,    25;
   7:  56,  504,  1080,   880,   300,    36;
   8:  84, 1008,  2970,  3520,  1950,   504,   49;
   9: 120, 1848,  7128, 11440,  9100,  3780,  784,   64;
  10: 165, 3168, 15444, 32032, 34125, 20160, 6664, 1152, 81;
%F Formula
The row polynomials satisfy (4*n-8+(2*n^3-8*n^2+8*n)*x)*P(n,x) = (4*n+4+(2*n^3-2*n^2+4)*x+(n^3-n)*x^2)*P(n-1,x) + (8*x+(4*n^2-4*n+2)*x^2+(n^2-n)*x^3)*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[((4*n + 4 + (2*n^3 - 2*n^2 + 4)*x + (n^3 - n)*x^2)*P[n - 1, x] + (8*x + (4*n^2 - 4*n + 2)*x^2 + (n^2 - n)*x^3)*D[P[n - 1, x], x])/(4*n - 8 + (2*n^3 - 8*n^2 + 8*n)*x)]];
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] = ((4*n + 4 + (2*n^3 - 2*n^2 + 4)*x + (n^3 - n)*x^2)*v[i - 1] + (8*x + (4*n^2 - 4*n + 2)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 1], x))/(4*n - 8 + (2*n^3 - 8*n^2 + 8*n)*x)); 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. [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