Real-rooted atlas

A253283 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(5 + 5\,x - 7\,n - 7\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(-9 - 27\,x - 27\,x^{2} - 9\,x^{3} + 9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3} - 2\,n^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-2}(x) + \left(16 + 62\,x + 46\,x^{2} - 16\,n - 64\,n x - 48\,n x^{2} + 4\,n^{2} + 16\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30257
403182546
50442132129307
605804209006812086
706135104036305910365314374
807210220511060288753792619825100108

All 15 rows

Technical details
Node id
n-e924d06af50baffa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b98f707c8be2e8fb

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 2,   5;
   4: 0, 3,  18,   25;
   5: 0, 4,  42,  132,   129;
   6: 0, 5,  80,  420,   900,    681;
   7: 0, 6, 135, 1040,  3630,   5910,   3653;
   8: 0, 7, 210, 2205, 11060,  28875,  37926,   19825;
   9: 0, 8, 308, 4200, 28210, 104440, 217644,  239624,  108545;
  10: 0, 9, 432, 7392, 63504, 311220, 911232, 1579536, 1497096, 598417;
%F Formula
The row polynomials satisfy (2*n^2-7*n+5+(2*n^2-7*n+5)*x)*P(n,x) = -(2*n^2-9*n+9+(6*n^2-27*n+27)*x+(6*n^2-27*n+27)*x^2+(2*n^2-9*n+9)*x^3)*P(n-2,x) + (4*n^2-16*n+16+(16*n^2-64*n+62)*x+(12*n^2-48*n+46)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9 + (6*n^2 - 27*n + 27)*x + (6*n^2 - 27*n + 27)*x^2 + (2*n^2 - 9*n + 9)*x^3)*P[n - 2, x] + (4*n^2 - 16*n + 16 + (16*n^2 - 64*n + 62)*x + (12*n^2 - 48*n + 46)*x^2)*P[n - 1, x])/(2*n^2 - 7*n + 5 + (2*n^2 - 7*n + 5)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9 + (6*n^2 - 27*n + 27)*x + (6*n^2 - 27*n + 27)*x^2 + (2*n^2 - 9*n + 9)*x^3)*v[i - 2] + (4*n^2 - 16*n + 16 + (16*n^2 - 64*n + 62)*x + (12*n^2 - 48*n + 46)*x^2)*v[i - 1])/(2*n^2 - 7*n + 5 + (2*n^2 - 7*n + 5)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.