Real-rooted atlas

A085880 → D → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(12 + 6\,x - 20\,n - 10\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x - 10\,n x + 4\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
38412
4607515150
54481064504562072
633601365011550294021031710
7253441671122257209900015840792533808
81921921996995405705327327307357358108130039798789

All 15 rows

Technical details
Node id
n-1f438888c60d8c6c · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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:        8,         4;
   4:       60,        75,         15;
   5:      448,      1064,        504,         56;
   6:     3360,     13650,      11550,       2940,       210;
   7:    25344,    167112,     225720,      99000,     15840,       792;
   8:   192192,   1996995,    4057053,    2732730,    735735,     81081,     3003;
   9:  1464320,  23554960,   69429360,   67507440,  27227200,   4964960,   400400,   11440;
  10: 11202048, 275894190, 1152366930, 1556647092, 883080198, 237080844, 31243212, 1925352, 43758;
%F Formula
The row polynomials satisfy (n^2-2*n)*P(n,x) = (8*n^2-20*n+12+(4*n^2-10*n+6)*x)*P(n-1,x) + (4*n^2-10*n+6)*x*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[((8*n^2 - 20*n + 12 + (4*n^2 - 10*n + 6)*x)*P[n - 1, x] + (4*n^2 - 10*n + 6)*x*D[P[n - 1, x], x])/(n^2 - 2*n)]];
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] = ((8*n^2 - 20*n + 12 + (4*n^2 - 10*n + 6)*x)*v[i - 1] + (4*n^2 - 10*n + 6)*x*deriv(v[i - 1], x))/(n^2 - 2*n)); 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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.