Real-rooted atlas

A089627 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - x + 6\,n - 3\,n x + n x^{2} + 4\,n^{2} x - 2\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-54 - 69\,x - 16\,x^{2} + 30\,n - 13\,n x - 47\,n x^{2} - 8\,n x^{3} + 26\,n^{2} x + 18\,n^{2} x^{2} - 4\,n^{2} x^{3} + 5\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-18 - 48\,x - 38\,x^{2} - 8\,x^{3} - 24\,n x - 62\,n x^{2} - 46\,n x^{3} - 8\,n x^{4} - 6\,n^{2} x^{2} - 14\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3347
471219
519481885
65116090301
7141540450801211
8393176418905604607

All 15 rows

Technical details
Node id
n-02da384cae3c4c62 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-975d9965822ff143

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
   3:    3,     4;
   4:    7,    12;
   5:   19,    48,    18;
   6:   51,   160,    90;
   7:  141,   540,   450,    80;
   8:  393,  1764,  1890,   560;
   9: 1107,  5712,  7560,  3360,  350;
  10: 3139, 18288, 28728, 16800, 3150;
%F Formula
The row polynomials satisfy (6*n-6+(4*n^2-3*n-1)*x+(n^3-2*n^2+n)*x^2)*P(n,x) = (30*n-54+(26*n^2-13*n-69)*x+(5*n^3+18*n^2-47*n-16)*x^2+(4*n^3-4*n^2-8*n)*x^3)*P(n-1,x) - (18+(24*n+48)*x+(6*n^2+62*n+38)*x^2+(14*n^2+46*n+8)*x^3+(8*n^2+8*n)*x^4)*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[((30*n - 54 + (26*n^2 - 13*n - 69)*x + (5*n^3 + 18*n^2 - 47*n - 16)*x^2 + (4*n^3 - 4*n^2 - 8*n)*x^3)*P[n - 1, x] - (18 + (24*n + 48)*x + (6*n^2 + 62*n + 38)*x^2 + (14*n^2 + 46*n + 8)*x^3 + (8*n^2 + 8*n)*x^4)*D[P[n - 1, x], x])/(6*n - 6 + (4*n^2 - 3*n - 1)*x + (n^3 - 2*n^2 + n)*x^2)]];
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] = ((30*n - 54 + (26*n^2 - 13*n - 69)*x + (5*n^3 + 18*n^2 - 47*n - 16)*x^2 + (4*n^3 - 4*n^2 - 8*n)*x^3)*v[i - 1] - (18 + (24*n + 48)*x + (6*n^2 + 62*n + 38)*x^2 + (14*n^2 + 46*n + 8)*x^3 + (8*n^2 + 8*n)*x^4)*deriv(v[i - 1], x))/(6*n - 6 + (4*n^2 - 3*n - 1)*x + (n^3 - 2*n^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, 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023