Real-rooted atlas

A116414 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 14\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-3 - 43\,x - 13\,x^{2} - x^{3} + 7\,n - 41\,n x - 13\,n x^{2} - n x^{3} + 15\,n^{2} + 3\,n^{2} x + 5\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-104 - 197\,x - 111\,x^{2} - 19\,x^{3} - x^{4} + 13\,n + 10\,n x + n x^{2} + 13\,n^{2} + 10\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1617
23212145
316499181282
48196922012411737
540124392178433830110557
6193632614813779364051036163477
79235214871796870331656460717421378324

All 15 rows

Technical details
Node id
n-4f92dbacbe46219f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c8a8416e5e397dda

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:       6,       1;
  2:      32,      12,       1;
  3:     164,      99,      18,       1;
  4:     819,     692,     201,      24,      1;
  5:    4012,    4392,    1784,     338,     30,      1;
  6:   19363,   26148,   13779,    3640,    510,     36,     1;
  7:   92352,  148717,   96870,   33165,   6460,    717,    42,    1;
  8:  436256,  817260,  636205,  269240,  67860,  10444,   959,   48,  1;
  9: 2044430, 4372455, 3968280, 2008110, 625170, 124299, 15792, 1236, 54, 1;
%F Formula
The row polynomials satisfy (2*n^3+2*n-14*n*x-2*n*x^2)*P(n,x) = (5*n^3+15*n^2+7*n-3+(n^3+3*n^2-41*n-43)*x-(13*n+13)*x^2-(n+1)*x^3)*P(n-1,x) + (13*n^2+13*n-104+(10*n^2+10*n-197)*x+(n^2+n-111)*x^2-19*x^3-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[((5*n^3 + 15*n^2 + 7*n - 3 + (n^3 + 3*n^2 - 41*n - 43)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (13*n^2 + 13*n - 104 + (10*n^2 + 10*n - 197)*x + (n^2 + n - 111)*x^2 - 19*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n - 14*n*x - 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] = ((5*n^3 + 15*n^2 + 7*n - 3 + (n^3 + 3*n^2 - 41*n - 43)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*v[i - 1] + (13*n^2 + 13*n - 104 + (10*n^2 + 10*n - 197)*x + (n^2 + n - 111)*x^2 - 19*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n - 14*n*x - 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023