Real-rooted atlas

A008459 → D → θ+1

palindromicγ-positivenonnegative coefficients

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

Recurrence

\(\left(-1 + 3\,n - 3\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(-n^{2} - 5\,n^{2} x + n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2448
3936954
41614414416320
525400900400251750
63690036003600900369072
749176411025196001102517644945276

All 15 rows

Technical details
Node id
n-3f18d778d359419c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b8a804edad022635

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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:  0;
  1:  1;
  2:  4,    4;
  3:  9,   36,     9;
  4: 16,  144,   144,     16;
  5: 25,  400,   900,    400,     25;
  6: 36,  900,  3600,   3600,    900,     36;
  7: 49, 1764, 11025,  19600,  11025,   1764,    49;
  8: 64, 3136, 28224,  78400,  78400,  28224,  3136,   64;
  9: 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81;
%F Formula
The row polynomials satisfy (n^3-3*n^2+3*n-1)*P(n,x) = (n^3-n^2+(3*n^3-5*n^2)*x)*P(n-1,x) + (2*n^2*x-2*n^2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 + (3*n^3 - 5*n^2)*x)*P[n - 1, x] + (2*n^2*x - 2*n^2*x^2)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 3*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 + (3*n^3 - 5*n^2)*x)*v[i - 1] + (2*n^2*x - 2*n^2*x^2)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 3*n - 1)); 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. [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