Real-rooted atlas

A063007 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 3\,x - 2\,x^{2} - 4\,n - 3\,n x + 5\,n x^{2} - 4\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(12 + 17\,x + 6\,x^{2} - 12\,n - 13\,n x + 3\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x - 12\,n^{2} x^{2} - 6\,n^{2} x^{3} + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 24\,x - 22\,x^{2} - 6\,x^{3} - 8\,n x^{2} - 12\,n x^{3} - 4\,n x^{4} + 4\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
313361867
46326427080677
53211800279016803506941
616831182024570224009450151271435
7898975852197820241920154350498966468735295
848639479248150368423083201975050964656252252274567559305

All 15 rows

Technical details
Node id
n-4967246d2d966d9c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f8d53c3083e7fcf9

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:       3,        4;
   3:      13,       36,       18;
   4:      63,      264,      270,        80;
   5:     321,     1800,     2790,      1680,       350;
   6:    1683,    11820,    24570,     22400,      9450,      1512;
   7:    8989,    75852,   197820,    241920,    154350,     49896,     6468;
   8:   48639,   479248,  1503684,   2308320,   1975050,    964656,   252252,    27456;
   9:  265729,  2994192, 10980900,  20291040,  21794850,  14270256,  5633628,  1235520,  115830;
  10: 1462563, 18548820, 77855580, 168241920, 217567350, 178810632, 94594500, 31299840, 5907330, 486200;
%F Formula
The row polynomials satisfy -(4*n-4+(3*n-3)*x-(n^3-4*n^2+5*n-2)*x^2)*P(n,x) = -(12*n-12+(4*n^2+13*n-17)*x-(3*n^3-12*n^2+3*n+6)*x^2-(2*n^3-6*n^2+4*n)*x^3)*P(n-1,x) - (8+24*x-(4*n^2-8*n-22)*x^2-(6*n^2-12*n-6)*x^3-(2*n^2-4*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[(-(12*n - 12 + (4*n^2 + 13*n - 17)*x - (3*n^3 - 12*n^2 + 3*n + 6)*x^2 - (2*n^3 - 6*n^2 + 4*n)*x^3)*P[n - 1, x] - (8 + 24*x - (4*n^2 - 8*n - 22)*x^2 - (6*n^2 - 12*n - 6)*x^3 - (2*n^2 - 4*n)*x^4)*D[P[n - 1, x], x])/(-4*n + 4 - (3*n - 3)*x + (n^3 - 4*n^2 + 5*n - 2)*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] = (-(12*n - 12 + (4*n^2 + 13*n - 17)*x - (3*n^3 - 12*n^2 + 3*n + 6)*x^2 - (2*n^3 - 6*n^2 + 4*n)*x^3)*v[i - 1] - (8 + 24*x - (4*n^2 - 8*n - 22)*x^2 - (6*n^2 - 12*n - 6)*x^3 - (2*n^2 - 4*n)*x^4)*deriv(v[i - 1], x))/(-4*n + 4 - (3*n - 3)*x + (n^3 - 4*n^2 + 5*n - 2)*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

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