Real-rooted atlas

A367270 → x ↦ x+1 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 - 16\,x - 32\,x^{2} + 5\,n + 40\,n x + 80\,n x^{2} - 2\,n^{2} - 16\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(10 + 16\,x - 24\,n - 32\,n x + 12\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
25813
3257248145
41295286722561585
5681360067205120128017401
63653236405808066560345606144192637
719825151704462000707840564480215040286722149561

All 15 rows

Technical details
Node id
n-2b39d1e9dbcbae64 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-7b26d207a1f3dfd1

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:      1;
  2:      5,       8;
  3:     25,      72,       48;
  4:    129,     528,      672,      256;
  5:    681,    3600,     6720,     5120,     1280;
  6:   3653,   23640,    58080,    66560,    34560,     6144;
  7:  19825,  151704,   462000,   707840,   564480,   215040,    28672;
  8: 108545,  958496,  3482304,  6684160,  7221760,  4300800,  1261568,  131072;
  9: 598417, 5988384, 25272576, 58318848, 79672320, 65028096, 30277632, 7077888, 589824;
%F Formula
The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(16*n^2-40*n+16)*x+(32*n^2-80*n+32)*x^2)*P(n-2,x) + (12*n^2-24*n+10+(16*n^2-32*n+16)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (16*n^2 - 40*n + 16)*x + (32*n^2 - 80*n + 32)*x^2)*P[n - 2, x] + (12*n^2 - 24*n + 10 + (16*n^2 - 32*n + 16)*x)*P[n - 1, x])/(2*n^2 - 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (16*n^2 - 40*n + 16)*x + (32*n^2 - 80*n + 32)*x^2)*v[i - 2] + (12*n^2 - 24*n + 10 + (16*n^2 - 32*n + 16)*x)*v[i - 1])/(2*n^2 - 3*n)); 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