Real-rooted atlas

A075497 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-13 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(-6 - 8\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(8 + 2\,x\right)\,P_{n-1}(x) + \left(2 + 2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
35530388
4384332724792
528413380117014057536
62250434022164403140240676352
71908633489222178335838071053787823488
81728064369551228341609894641691201428056089431168
9166407854066135236987492160159443578190423864262927929114334720
101698110804658610024899751202541175607077504010967292952560452401080101462505984
11182991232755605496546633272943401479364013473953902608701482962205419654807375514301119678456832
1220757687872691167640049212904280863786257084251433336005885391336835440144723779763785760115060184812277730197504
132471428728098938488950041315554073782102563422046846545007151512847815208822082064004239054244016299726368897401730302340134100750954496
143080290246120120134443057741933456532935216759649095644861767522616027506971351865587538596647326490714462363180883430326901195594425225229121463194931666944
Technical details
Node id
n-9c44db94ed4c4357 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-60207baa81022973

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:        8,        2;
  3:       55,       30,        3;
  4:      384,      332,       72,        4;
  5:     2841,     3380,     1170,      140,       5;
  6:    22504,    34022,    16440,     3140,     240,      6;
  7:   190863,   348922,   217833,    58380,    7105,    378,     7;
  8:  1728064,  3695512,  2834160,   989464,  169120,  14280,   560,   8;
  9: 16640785, 40661352, 36987492, 16015944, 3578190, 423864, 26292, 792, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(13+6*x+x^2)*P(n-2,x) - (6+8*x+2*x^2)*d/dx P(n-2,x) + (8+2*x)*P(n-1,x) + (2+2*x)*d/dx 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[-(13 + 6*x + x^2)*P[n - 2, x] - (6 + 8*x + 2*x^2)*D[P[n - 2, x], x] + (8 + 2*x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];
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] = -(13 + 6*x + x^2)*v[i - 2] - (6 + 8*x + 2*x^2)*deriv(v[i - 2], x) + (8 + 2*x)*v[i - 1] + (2 + 2*x)*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 - 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.