Real-rooted atlas

A386789 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-1 - x + 5\,x^{2} + 9\,x^{3} + 4\,x^{4} + n + 3\,n x - n x^{2} - 7\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 18\,x^{3} + 14\,x^{4} + 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
203912
304284476
40560210225500
506108624145211823372
607175147055309625632123128
708264300016120447606218434232160568

All 15 rows

Technical details
Node id
n-cd8b20fbf5376fcd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cb8a225300b7095e

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: 0,  2;
  2: 0,  3,   9;
  3: 0,  4,  28,   44;
  4: 0,  5,  60,  210,   225;
  5: 0,  6, 108,  624,  1452,   1182;
  6: 0,  7, 175, 1470,  5530,   9625,    6321;
  7: 0,  8, 264, 3000, 16120,  44760,   62184,   34232;
  8: 0,  9, 378, 5544, 39690, 155610,  341712,  394884,  187137;
  9: 0, 10, 520, 9520, 86800, 448840, 1380400, 2504320, 2476840, 1030490;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n-1+(3*n-1)*x-(n-5)*x^2-(7*n-9)*x^3-(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*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 - 1 + (3*n - 1)*x - (n - 5)*x^2 - (7*n - 9)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + 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 - 1 + (3*n - 1)*x - (n - 5)*x^2 - (7*n - 9)*x^3 - (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + 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

Same fingerprint ?

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.