Real-rooted atlas

chebyshev_u → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21116431
32468488148
44520827612816673
5765181088928320322962
611911203384448028007686412735
71762184892816720160007872179212853800

All 15 rows

Technical details
Node id
n-ad64eb2286c988ad · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-cd68f890aaf2d75a

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:   4,    2;
  2:  11,   16,     4;
  3:  24,   68,    48,      8;
  4:  45,  208,   276,    128,     16;
  5:  76,  518,  1088,    928,    320,     32;
  6: 119, 1120,  3384,   4480,   2800,    768,     64;
  7: 176, 2184,  8928,  16720,  16000,   7872,   1792,   128;
  8: 249, 3936, 20856,  52096,  69680,  51968,  21056,  4096,  256;
  9: 340, 6666, 44352, 141856, 250432, 258720, 157696, 54272, 9216, 512;
%F Formula
The row polynomials satisfy (n^3+2*n+n*x-n*x^2)*P(n,x) = (n^3+3*n^2+5*n+3+(n^3+3*n^2+4*n+2)*x-(n+1)*x^2-(n+1)*x^3)*P(n-1,x) + ((2*n^2+2*n+2)*x+(n^2+n-1)*x^2-3*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[((n^3 + 3*n^2 + 5*n + 3 + (n^3 + 3*n^2 + 4*n + 2)*x - (n + 1)*x^2 - (n + 1)*x^3)*P[n - 1, x] + ((2*n^2 + 2*n + 2)*x + (n^2 + n - 1)*x^2 - 3*x^3 - x^4)*D[P[n - 1, x], x])/(n^3 + 2*n + n*x - 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] = ((n^3 + 3*n^2 + 5*n + 3 + (n^3 + 3*n^2 + 4*n + 2)*x - (n + 1)*x^2 - (n + 1)*x^3)*v[i - 1] + ((2*n^2 + 2*n + 2)*x + (n^2 + n - 1)*x^2 - 3*x^3 - x^4)*deriv(v[i - 1], x))/(n^3 + 2*n + n*x - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023