Real-rooted atlas

A371081 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 - 8\,x - x^{2} - 60\,n - 8\,n x - 37\,n^{2} - 2\,n^{2} x - 10\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-28\,x - 6\,x^{2} - 36\,n x - 2\,n x^{2} - 15\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - x^{3} - 4\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(20 + 2\,x + 12\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(9\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
252254
3294823233183
4203072246886484228412
51715443228749441154401440520146261
6176060160038217708820935776397280278062164114530
7216690624000585547038724111411776105781248112210048727279463647875

All 15 rows

Technical details
Node id
n-4a391bbf6eb1853e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-98924351874de76b

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 -1)
  • 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:                0;
  1:                1;
  2:               52,                2;
  3:             2948,              232,               3;
  4:           203072,            24688,             648,             4;
  5:         17154432,          2874944,          115440,          1440,            5;
  6:       1760601600,        382177088,        20935776,        397280,         2780,          6;
  7:     216690624000,      58554703872,      4111411776,     105781248,      1122100,       4872,       7;
  8:   31569232896000,   10329126202368,    893668763904,   29282663168,    419484160,    2754528,    7952,     8;
  9: 5379460107264000, 2087647201456128, 216569161181952, 8687920122880, 157834251200, 1395848064, 6084288, 12288, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4+10*n^3+37*n^2+60*n+36+(2*n^2+8*n+8)*x+x^2)*P(n-2,x) - ((2*n^3+15*n^2+36*n+28)*x+(2*n+6)*x^2)*d/dx P(n-2,x) - ((n^2+4*n+4)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+12*n+20+2*x)*P(n-1,x) + (2*n+9)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 + (2*n^2 + 8*n + 8)*x + x^2)*P[n - 2, x] - ((2*n^3 + 15*n^2 + 36*n + 28)*x + (2*n + 6)*x^2)*D[P[n - 2, x], x] - ((n^2 + 4*n + 4)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 12*n + 20 + 2*x)*P[n - 1, x] + (2*n + 9)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 + (2*n^2 + 8*n + 8)*x + x^2)*v[i - 2] - ((2*n^3 + 15*n^2 + 36*n + 28)*x + (2*n + 6)*x^2)*deriv(v[i - 2], x) - ((n^2 + 4*n + 4)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 12*n + 20 + 2*x)*v[i - 1] + (2*n + 9)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.