Real-rooted atlas

A172431 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(10\,n - 2\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(12 - 4\,x^{2} - x^{3} + 16\,n + 2\,n x - 4\,n x^{2} - n x^{3} + 6\,n^{2} + 3\,n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 10\,x^{2} - 7\,x^{3} - x^{4} + 4\,n x + n x^{2} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
276114
314229146
4256245121145
54114716476151444
6633084863401151811331
792588124212106101622113926

All 15 rows

Technical details
Node id
n-9e2a798673fae340 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-16b10bb8625df23c

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:   3,    1;
  2:   7,    6,    1;
  3:  14,   22,    9,    1;
  4:  25,   62,   45,   12,    1;
  5:  41,  147,  164,   76,   15,    1;
  6:  63,  308,  486,  340,  115,   18,    1;
  7:  92,  588, 1242, 1210,  610,  162,   21,   1;
  8: 129, 1044, 2838, 3652, 2535,  994,  217,  24,  1;
  9: 175, 1749, 5940, 9724, 8827, 4725, 1512, 280, 27, 1;
%F Formula
The row polynomials satisfy (2*n^3+10*n-2*n*x-2*n*x^2)*P(n,x) = (2*n^3+6*n^2+16*n+12+(n^3+3*n^2+2*n)*x-(4*n+4)*x^2-(n+1)*x^3)*P(n-1,x) + ((4*n^2+4*n+8)*x+(n^2+n-10)*x^2-7*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 6*n^2 + 16*n + 12 + (n^3 + 3*n^2 + 2*n)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*P[n - 1, x] + ((4*n^2 + 4*n + 8)*x + (n^2 + n - 10)*x^2 - 7*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 10*n - 2*n*x - 2*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 6*n^2 + 16*n + 12 + (n^3 + 3*n^2 + 2*n)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*v[i - 1] + ((4*n^2 + 4*n + 8)*x + (n^2 + n - 10)*x^2 - 7*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 10*n - 2*n*x - 2*n*x^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

Same fingerprint ?

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