Real-rooted atlas

A358735 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-6\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 12\,n - 8\,n x + 2\,n x^{2} - 6\,n^{2} + 12\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 10\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 8\,x + 6\,x^{2}\)
\(P_{3}(x) = 1 + 20\,x + 48\,x^{2} + 24\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
218615
3120482493
4140210304120675
5170672198421407205587
61112176490401904016848504051845
7116840323256011504019291214767240320532705

All 15 rows

Technical details
Node id
n-67fe0ee394103768 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f387468580877db5

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: 1,   2;
  2: 1,   8,     6;
  3: 1,  20,    48,     24;
  4: 1,  40,   210,    304,     120;
  5: 1,  70,   672,   1984,    2140,     720;
  6: 1, 112,  1764,   9040,   19040,   16848,     5040;
  7: 1, 168,  4032,  32560,  115040,  192912,   147672,    40320;
  8: 1, 240,  8316,  99088,  536640, 1473360,  2090816,  1430784,   362880;
  9: 1, 330, 15840, 265408, 2072980, 8549520, 19504240, 24326784, 15212448, 3628800;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 8*x + 6*x^2, P(3,x) = 1 + 20*x + 48*x^2 + 24*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 8*x + 6*x^2;
P[3, x_] := 1 + 20*x + 48*x^2 + 24*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
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 + 2*x; v[3] = 1 + 8*x + 6*x^2; v[4] = 1 + 20*x + 48*x^2 + 24*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 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. [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