Real-rooted atlas

A327997 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-54 + 45\,n - 9\,n^{2}\right)\,P_{n-4}(x) + \left(-6 - 12\,x - 9\,n + 6\,n x + 6\,n^{2}\right)\,P_{n-3}(x) + \left(8 - 2\,x - x^{2} - 8\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 9 + 14\,x + 3\,x^{2}\)
\(P_{3}(x) = 27 + 76\,x + 36\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2914326
32776364143
481384303725845
52431938226586012565437
6729102301648288202000198738466
7218758644122913858762677540742948300771

All 15 rows

Technical details
Node id
n-40abbfa038458015 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b76628d7fb7fc96d

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 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;
  2:     9,      14,       3;
  3:    27,      76,      36,       4;
  4:    81,     384,     303,      72,       5;
  5:   243,    1938,    2265,     860,     125,       6;
  6:   729,   10230,   16482,    8820,    2000,     198,      7;
  7:  2187,   58644,  122913,   85876,   26775,    4074,    294,     8;
  8:  6561,  375984,  970974,  837424,  333095,   68712,   7546,   416,   9;
  9: 19683, 2741490, 8294778, 8445272, 4068225, 1057770, 156114, 13008, 567, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-9*n-6+(6*n-12)*x)*P(n-3,x) - (n^2+8*n-8+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 9 + 14*x + 3*x^2, P(3,x) = 27 + 76*x + 36*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 9 + 14*x + 3*x^2;
P[3, x_] := 27 + 76*x + 36*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 9 + 14*x + 3*x^2; v[4] = 27 + 76*x + 36*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*v[i - 3] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*x)*v[i - 1]); 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. [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