Real-rooted atlas

A130534 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9\,x^{2} - 6\,n x - n^{2}\right)\,P_{n-2}(x) + \left(1 + 6\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
22133963
310224016227531
458217401773702814878
53852135721777510665283524348942
6291241159561808011405355670010935729534780
72481121087272193208418015489468902772634082421876336180

All 15 rows

Technical details
Node id
n-867c747f4f1fd091 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-16a146ae3dc3b67e

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:        5,         3;
  2:       21,        33,         9;
  3:      102,       240,       162,        27;
  4:      582,      1740,      1773,       702,        81;
  5:     3852,     13572,     17775,     10665,      2835,       243;
  6:    29124,    115956,    180801,    140535,     56700,     10935,      729;
  7:   248112,   1087272,   1932084,   1801548,    946890,    277263,    40824,    2187;
  8:  2353968,  11153160,  21969792,  23502528,  14904729,   5708556,  1275750,  148716,   6561;
  9: 24628320, 124566768, 266900508, 318323412, 232834581, 108086643, 31689630, 5603094, 531441, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+6*n*x+9*x^2)*P(n-2,x) + (2*n+1+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n*x + 9*x^2)*P[n - 2, x] + (2*n + 1 + 6*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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n*x + 9*x^2)*v[i - 2] + (2*n + 1 + 6*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.