Real-rooted atlas

A114656 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 2\,n + 2\,n x - n x^{2} + 2\,n^{2} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 2\,x^{3} - 10\,n - 8\,n x - 11\,n x^{2} - 3\,n x^{3} + 8\,n^{2} x + 6\,n^{2} x^{2} - 2\,n^{2} x^{3} + 5\,n^{3} x^{2} + 3\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2 + x^{2} + x^{4} + 4\,n x + n x^{2} - 4\,n x^{3} - n x^{4} + 2\,n^{2} x^{2} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31116330
445102454196
51976084389651344
690335303570132017569504
742792025626505140803225288768640
8207931155981860391300604357568464418503360

All 15 rows

Technical details
Node id
n-ab3980fcc888883c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-78ed572c24a1b447

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,       2;
   3:     11,      16,       3;
   4:     45,     102,      45,       4;
   5:    197,     608,     438,      96,       5;
   6:    903,    3530,    3570,    1320,     175,       6;
   7:   4279,   20256,   26505,   14080,    3225,     288,      7;
   8:  20793,  115598,  186039,  130060,   43575,    6846,    441,     8;
   9: 103049,  658048, 1258572, 1096256,  490070,  114240,  13132,   640,   9;
  10: 518859, 3742290, 8298180, 8677200, 4885650, 1536948, 265188, 23328, 891, 10;
%F Formula
The row polynomials satisfy -(2*n+2-(2*n^2+2*n)*x-(n^3-n)*x^2)*P(n,x) = -(10*n-2-(8*n^2-8*n+8)*x-(5*n^3+6*n^2-11*n)*x^2-(3*n^3-2*n^2-3*n+2)*x^3)*P(n-1,x) - (2-4*n*x-(2*n^2+n+1)*x^2+4*n*x^3+(2*n^2+n-1)*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[(-(10*n - 2 - (8*n^2 - 8*n + 8)*x - (5*n^3 + 6*n^2 - 11*n)*x^2 - (3*n^3 - 2*n^2 - 3*n + 2)*x^3)*P[n - 1, x] - (2 - 4*n*x - (2*n^2 + n + 1)*x^2 + 4*n*x^3 + (2*n^2 + n - 1)*x^4)*D[P[n - 1, x], x])/(-2*n - 2 + (2*n^2 + 2*n)*x + (n^3 - 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] = (-(10*n - 2 - (8*n^2 - 8*n + 8)*x - (5*n^3 + 6*n^2 - 11*n)*x^2 - (3*n^3 - 2*n^2 - 3*n + 2)*x^3)*v[i - 1] - (2 - 4*n*x - (2*n^2 + n + 1)*x^2 + 4*n*x^3 + (2*n^2 + n - 1)*x^4)*deriv(v[i - 1], x))/(-2*n - 2 + (2*n^2 + 2*n)*x + (n^3 - 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