Real-rooted atlas

A281260 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 4\,n x + 2\,n x^{2} - n^{2} + 2\,n^{2} x + n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(x - 2\,x^{2} + x^{3} - 2\,n - 2\,n x + 2\,n x^{3} - n^{2} - 8\,n^{2} x + 2\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} + 6\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-x + x^{2} + x^{3} - x^{4} - 2\,n x + 4\,n x^{2} - 2\,n x^{4} + 4\,n^{2} x - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
228313
321820444
423275405154
5250204220706552
627245584052511272013
729888825202660109216887436
82128157563841029070562058240927742
92162260014280327603410416464360033010104312
102200405929040900901330569702034848594044011394706
112242606054780221100440748454608245520683109350572121501304
12228887239724049549512835681783782135907256628012584014157728135735188
13233812180164164103103033693666078072622908036525061211210220220207489101421989840
1423921657526572020170158120112184654472453880019324305899756024334313690962957511201584579165
Technical details
Node id
n-0987851239956860 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e4482831ac6cbc49

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: 2,   2;
  2: 2,   8,    3;
  3: 2,  18,   20,     4;
  4: 2,  32,   75,    40,     5;
  5: 2,  50,  204,   220,    70,     6;
  6: 2,  72,  455,   840,   525,   112,     7;
  7: 2,  98,  888,  2520,  2660,  1092,   168,    8;
  8: 2, 128, 1575,  6384, 10290,  7056,  2058,  240,   9;
  9: 2, 162, 2600, 14280, 32760, 34104, 16464, 3600, 330, 10;
%F Formula
The row polynomials satisfy (n^4+2*n^3-n^2-2*n+(2*n^3+2*n^2-4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^4+2*n^3-n^2-2*n+(3*n^4+6*n^3-8*n^2-2*n+1)*x+(6*n^3+2*n^2-2)*x^2+(3*n^2+2*n+1)*x^3)*P(n-1,x) + ((2*n^3+4*n^2-2*n-1)*x-(2*n^3-4*n-1)*x^2-(4*n^2-1)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 6*n^3 - 8*n^2 - 2*n + 1)*x + (6*n^3 + 2*n^2 - 2)*x^2 + (3*n^2 + 2*n + 1)*x^3)*P[n - 1, x] + ((2*n^3 + 4*n^2 - 2*n - 1)*x - (2*n^3 - 4*n - 1)*x^2 - (4*n^2 - 1)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - n^2 - 2*n + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 6*n^3 - 8*n^2 - 2*n + 1)*x + (6*n^3 + 2*n^2 - 2)*x^2 + (3*n^2 + 2*n + 1)*x^3)*v[i - 1] + ((2*n^3 + 4*n^2 - 2*n - 1)*x - (2*n^3 - 4*n - 1)*x^2 - (4*n^2 - 1)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - n^2 - 2*n + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*n)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.