Real-rooted atlas

A089503 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 - 6\,x^{2} - 8\,x^{3} - 3\,x^{4} + 4\,n x + 10\,n x^{2} + 6\,n x^{3} - 2\,n x^{4} - 2\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(-x^{2} - 3\,x^{3} - 3\,x^{4} - x^{5} - n x^{3} - 3\,n x^{4} - 3\,n x^{5} - n x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2156
31144358
4127219529776
51446664084850113296
616515701753090005169021278192
7190316555460508635230454640104556882352
81119573314451520518751635146167457719110676833196688256

All 15 rows

Technical details
Node id
n-193bc293afe2c18c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d37e58d0d0ef7fc3

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: 1,   5;
   3: 1,  14,    43;
   4: 1,  27,   219,    529;
   5: 1,  44,   666,   4084,     8501;
   6: 1,  65,  1570,  17530,    90005,    169021;
   7: 1,  90,  3165,  55460,   508635,   2304546,    4010455;
   8: 1, 119,  5733, 144515,  2051875,  16351461,   67457719,   110676833;
   9: 1, 152,  9604, 328664,  6645310,  81058376,  580699252,  2226361544,  3484717129;
  10: 1, 189, 15156, 675444, 18396126, 316786806, 3440379684, 22659789636, 81889548489, 123320412181;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+4*n*x+(n^2+10*n-6)*x^2+(3*n^2+6*n-8)*x^3+(3*n^2-2*n-3)*x^4+(n^2-2*n)*x^5)*P(n-1,x) - (x^2+(n+3)*x^3+(3*n+3)*x^4+(3*n+1)*x^5+n*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 4*n*x + (n^2 + 10*n - 6)*x^2 + (3*n^2 + 6*n - 8)*x^3 + (3*n^2 - 2*n - 3)*x^4 + (n^2 - 2*n)*x^5)*P[n - 1, x] - (x^2 + (n + 3)*x^3 + (3*n + 3)*x^4 + (3*n + 1)*x^5 + n*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 4*n*x + (n^2 + 10*n - 6)*x^2 + (3*n^2 + 6*n - 8)*x^3 + (3*n^2 - 2*n - 3)*x^4 + (n^2 - 2*n)*x^5)*v[i - 1] - (x^2 + (n + 3)*x^3 + (3*n + 3)*x^4 + (3*n + 1)*x^5 + n*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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

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.