Real-rooted atlas

A091044 → D → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(1 + 3\,x + 3\,n - 15\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
266
3562480
425236060672
5792237613201204608
620021029612012364021028160
743683432068640436808400336159744

All 15 rows

Technical details
Node id
n-a7bdba1fe4a3fca8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c8712d72043b8477

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:     0;
  2:     6;
  3:    56,     24;
  4:   252,    360,      60;
  5:   792,   2376,    1320,     120;
  6:  2002,  10296,   12012,    3640,     210;
  7:  4368,  34320,   68640,   43680,    8400,    336;
  8:  8568,  95472,  291720,  318240,  128520,  17136,   504;
  9: 15504, 232560, 1007760, 1679600, 1162800, 325584, 31920, 720;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2+3*n+1+(6*n^2-15*n+3)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (6*n^2 - 15*n + 3)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
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] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n + 1 + (6*n^2 - 15*n + 3)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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