Real-rooted atlas

A131689 → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-2 - 4\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(-2 - 5\,x - 4\,x^{2} - x^{3}\right)\,P_{n-2}'(x) + \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3549
431481897
5233532396961257
6207162406930336060019201
7213057968411754085440306004320338889
82492711113168204661819824001066800302400352806795937

All 15 rows

Technical details
Node id
n-999d2dacd9cc0a1f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-59373de250012a5b

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:        5,         4;
   4:       31,        48,        18;
   5:      233,       532,       396,         96;
   6:     2071,      6240,      6930,       3360,       600;
   7:    21305,     79684,    117540,      85440,     30600,      4320;
   8:   249271,   1113168,   2046618,    1982400,   1066800,    302400,     35280;
   9:  3270713,  16975732,  37369836,   45220896,  32482800,  13849920,   3245760,   322560;
  10: 47580151, 281392320, 721414530, 1047039840, 940854600, 535973760, 189030240, 37739520, 3265920;
%F Formula
The row polynomials satisfy P(n,x) = -(2+4*x+2*x^2)*P(n-2,x) - (2+5*x+4*x^2+x^3)*d/dx P(n-2,x) + (5+4*x)*P(n-1,x) + (2+3*x+x^2)*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[-(2 + 4*x + 2*x^2)*P[n - 2, x] - (2 + 5*x + 4*x^2 + x^3)*D[P[n - 2, x], x] + (5 + 4*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], 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] = -(2 + 4*x + 2*x^2)*v[i - 2] - (2 + 5*x + 4*x^2 + x^3)*deriv(v[i - 2], x) + (5 + 4*x)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.