Real-rooted atlas

A054142 → D → D

nonnegative coefficients

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

Recurrence

\(\left(10 - 9\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(3 + 12\,x - 5\,n - 21\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} - n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
312618
4306012102
55621018020466
690504840420301884
713299025202520840427044
8182171659409240630015125624946
924027301201225740277201386025207284894
1030640802184060060900907207227720396090280218
11380581436720123760240240270270168168514805940110902882
124627980581402325605569208168167207203603609009085801322852760
1355210626877804069801162800211629624504481750320720720150150120121568868840
14650138001275126729802238390488376070543206651216393822013613602402401638018227199010
Technical details
Node id
n-bfc760761568f51d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-58897385a6234e7d

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:   2;
  3:  12,    6;
  4:  30,   60,    12;
  5:  56,  210,   180,    20;
  6:  90,  504,   840,   420,    30;
  7: 132,  990,  2520,  2520,   840,    42;
  8: 182, 1716,  5940,  9240,  6300,  1512,   56;
  9: 240, 2730, 12012, 25740, 27720, 13860, 2520, 72;
%F Formula
The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (2*n^2-5*n+3+(6*n^2-21*n+12)*x)*P(n-1,x) - ((n-1)*x+(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 + (6*n^2 - 21*n + 12)*x)*P[n - 1, x] - ((n - 1)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];
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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 3 + (6*n^2 - 21*n + 12)*x)*v[i - 1] - ((n - 1)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); 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