Real-rooted atlas

A075498 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 10\,x + 10\,x^{2} + 3\,x^{3} - 3\,n x - 6\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
29271955
327144225109505
481729192620197425497
52433726149042537119764581569823
672919683111861274644339921209376511931007407
72187107892836163277676148550594696182238554349811816157905

All 15 rows

Technical details
Node id
n-f656951313aec62a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-90c6c54322d6b2c6

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:     3,       4;
  2:     9,      27,       19;
  3:    27,     144,      225,       109;
  4:    81,     729,     1926,      2019,       742;
  5:   243,    3726,    14904,     25371,     19764,       5815;
  6:   729,   19683,   111861,    274644,    339921,     209376,      51193;
  7:  2187,  107892,   836163,   2776761,   4855059,    4696182,    2385543,     498118;
  8:  6561,  610173,  6292728,  27198261,  63060201,   85195557,   67451238,   29103057,   5296321;
  9: 19683, 3529818, 47934666, 263223675, 778237389, 1377786915, 1511709156, 1010933163, 378560493, 60987817;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3-(3*n-10)*x-(6*n-10)*x^2-(3*n-3)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*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[((3 - (3*n - 10)*x - (6*n - 10)*x^2 - (3*n - 3)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = ((3 - (3*n - 10)*x - (6*n - 10)*x^2 - (3*n - 3)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(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.