Real-rooted atlas

A033282 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 6\,n x - 2\,n x^{2} + n^{2} + 3\,n^{2} x - n^{2} x^{2} + 3\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(8\,x^{2} - 4\,x^{3} + 2\,n + 18\,n x + 10\,n x^{2} - 6\,n x^{3} + 7\,n^{2} + 25\,n^{2} x + 3\,n^{2} x^{2} - 2\,n^{2} x^{3} + 5\,n^{3} + 12\,n^{3} x + n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 2\,x^{4} + 4\,n x + 6\,n x^{2} - 2\,n x^{4} + 6\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
211401566
31914423156450
429364117611762102955
54176039607800561079218963
6551404105753300046035257403003119812
77123802425510780023523525225211511511440748548

All 15 rows

Technical details
Node id
n-5ef5d1e2cb1d41ee · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2fb8ba1337b01a08

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:   5,    4;
  2:  11,   40,    15;
  3:  19,  144,   231,     56;
  4:  29,  364,  1176,   1176,     210;
  5:  41,  760,  3960,   7800,    5610,     792;
  6:  55, 1404, 10575,  33000,   46035,   25740,    3003;
  7:  71, 2380, 24255, 107800,  235235,  252252,  115115,   11440;
  8:  89, 3784, 49896, 296296,  910910, 1513512, 1313312,  505648,   43758;
  9: 109, 5724, 94536, 718536, 2923830, 6787872, 9059232, 6587568, 2191878, 167960;
%F Formula
The row polynomials satisfy (n^4+3*n^3+n^2-2*n+(3*n^2+6*n)*x-(n^2+2*n)*x^2)*P(n,x) = (n^4+5*n^3+7*n^2+2*n+(2*n^4+12*n^3+25*n^2+18*n)*x+(3*n^2+10*n+8)*x^2-(2*n^2+6*n+4)*x^3)*P(n-1,x) + ((2*n^3+6*n^2+4*n)*x+(2*n^3+6*n^2+6*n+2)*x^2-(2*n+2)*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[((n^4 + 5*n^3 + 7*n^2 + 2*n + (2*n^4 + 12*n^3 + 25*n^2 + 18*n)*x + (3*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 4)*x^3)*P[n - 1, x] + ((2*n^3 + 6*n^2 + 4*n)*x + (2*n^3 + 6*n^2 + 6*n + 2)*x^2 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 3*n^3 + n^2 - 2*n + (3*n^2 + 6*n)*x - (n^2 + 2*n)*x^2)]];
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] = ((n^4 + 5*n^3 + 7*n^2 + 2*n + (2*n^4 + 12*n^3 + 25*n^2 + 18*n)*x + (3*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 4)*x^3)*v[i - 1] + ((2*n^3 + 6*n^2 + 4*n)*x + (2*n^3 + 6*n^2 + 6*n + 2)*x^2 - (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 3*n^3 + n^2 - 2*n + (3*n^2 + 6*n)*x - (n^2 + 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 - 1, v[i])))};
T(9)

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. [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