Real-rooted atlas

A121314 → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 18\,x - 10\,n + 42\,n x + 4\,n^{2} - 32\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(6 + 26\,x + 8\,x^{2} - 10\,n - 26\,n x + 46\,n x^{2} + 4\,n^{2} - 8\,n^{2} x - 72\,n^{2} x^{2} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 9\,x^{2} + 4\,x^{3} - 2\,n x - 4\,n x^{2} + 16\,n x^{3} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2123
316310
411018433
511445405105
611884140756324
71221353363501267977

All 15 rows

Technical details
Node id
n-bfb3b94361638ad4 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f86d77f8169a378b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0;
  1: 1;
  2: 1,  2;
  3: 1,  6,   3;
  4: 1, 10,  18,    4;
  5: 1, 14,  45,   40,    5;
  6: 1, 18,  84,  140,   75,    6;
  7: 1, 22, 135,  336,  350,  126,    7;
  8: 1, 26, 198,  660, 1050,  756,  196,   8;
  9: 1, 30, 273, 1144, 2475, 2772, 1470, 288, 9;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6+(8*n^3-32*n^2+42*n-18)*x)*P(n,x) = (4*n^2-10*n+6+(8*n^3-8*n^2-26*n+26)*x+(24*n^3-72*n^2+46*n+8)*x^2)*P(n-1,x) - ((2*n-2)*x+(4*n^2+4*n-9)*x^2+(16*n^2-16*n-4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 6 + (8*n^3 - 8*n^2 - 26*n + 26)*x + (24*n^3 - 72*n^2 + 46*n + 8)*x^2)*P[n - 1, x] - ((2*n - 2)*x + (4*n^2 + 4*n - 9)*x^2 + (16*n^2 - 16*n - 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (8*n^3 - 32*n^2 + 42*n - 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 10*n + 6 + (8*n^3 - 8*n^2 - 26*n + 26)*x + (24*n^3 - 72*n^2 + 46*n + 8)*x^2)*v[i - 1] - ((2*n - 2)*x + (4*n^2 + 4*n - 9)*x^2 + (16*n^2 - 16*n - 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (8*n^3 - 32*n^2 + 42*n - 18)*x)); 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

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