Real-rooted atlas

A049218 → D+1

Proved for all n all roots realinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(2\,n - 5\,n^{2} + 4\,n^{3} - n^{4}\right)\,P_{n-4}(x) + \left(-2\,n x + 2\,n^{2} x\right)\,P_{n-3}(x) + \left(-x^{2} - 2\,n^{2}\right)\,P_{n-2}(x) + \left(2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = -2 + 2\,x + x^{2}\)
\(P_{2}(x) = -8 - 8\,x + 3\,x^{2} + x^{3}\)
\(P_{3}(x) = 24 - 40\,x - 20\,x^{2} + 4\,x^{3} + x^{4}\)
\(P_{4}(x) = 184 + 184\,x - 120\,x^{2} - 40\,x^{3} + 5\,x^{4} + x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1-2211
2-8-831-12
324-40-2041-31
4184184-120-4051214
5-7201568784-280-70611289
6-8448-844873922464-560-11271-7704
740320-104704-52352255366384-1008-16881-85983
8648576648576-689280-2297607224014448-1680-24091462890

All 15 rows

Technical details
Node id
n-841c96f8d5e729f0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6d9f0e6f91576231

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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:        -2,         2,        1;
   2:        -8,        -8,        3,        1;
   3:        24,       -40,      -20,        4,         1;
   4:       184,       184,     -120,      -40,         5,        1;
   5:      -720,      1568,      784,     -280,       -70,        6,      1;
   6:     -8448,     -8448,     7392,     2464,      -560,     -112,      7,     1;
   7:     40320,   -104704,   -52352,    25536,      6384,    -1008,   -168,     8,     1;
   8:    648576,    648576,  -689280,  -229760,     72240,    14448,  -1680,  -240,     9,    1;
   9:  -3628800,  10720512,  5360256, -3217280,   -804320,   177408,  29568, -2640,  -330,   10,  1;
  10: -74972160, -74972160, 91901568, 30633856, -11968000, -2393600, 391776, 55968, -3960, -440, 11, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-4*n^3+5*n^2-2*n)*P(n-4,x) + (2*n^2-2*n)*x*P(n-3,x) - (2*n^2+x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 5, with P(1,x) = -2 + 2*x + x^2, P(2,x) = -8 - 8*x + 3*x^2 + x^3, P(3,x) = 24 - 40*x - 20*x^2 + 4*x^3 + x^4, P(4,x) = 184 + 184*x - 120*x^2 - 40*x^3 + 5*x^4 + x^5.
%t Mathematica
P[1, x_] := -2 + 2*x + x^2;
P[2, x_] := -8 - 8*x + 3*x^2 + x^3;
P[3, x_] := 24 - 40*x - 20*x^2 + 4*x^3 + x^4;
P[4, x_] := 184 + 184*x - 120*x^2 - 40*x^3 + 5*x^4 + x^5;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 4*n^3 + 5*n^2 - 2*n)*P[n - 4, x] + (2*n^2 - 2*n)*x*P[n - 3, x] - (2*n^2 + x^2)*P[n - 2, x] + 2*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = -2 + 2*x + x^2; v[2] = -8 - 8*x + 3*x^2 + x^3; v[3] = 24 - 40*x - 20*x^2 + 4*x^3 + x^4; v[4] = 184 + 184*x - 120*x^2 - 40*x^3 + 5*x^4 + x^5; for(n = 5, N, my(i = n); v[i] = -(n^4 - 4*n^3 + 5*n^2 - 2*n)*v[i - 4] + (2*n^2 - 2*n)*x*v[i - 3] - (2*n^2 + x^2)*v[i - 2] + 2*x*v[i - 1]); 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

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