Real-rooted atlas

A229032 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7\,n - 6\,n x + 8\,n x^{2} + 2\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(9 + 6\,x - 32\,x^{2} - 32\,x^{3} - 33\,n - 50\,n x + 16\,n x^{2} + 32\,n x^{3} + 8\,n^{2} + 40\,n^{2} x + 32\,n^{2} x^{2} + 10\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(-18 - 102\,x - 212\,x^{2} - 192\,x^{3} - 64\,x^{4} - 36\,n - 132\,n x - 160\,n x^{2} - 64\,n x^{3} - 12\,n^{2} - 28\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
211415
3361652
413310416253
5454464961014
615512060720644395
751928416396851218088

All 15 rows

Technical details
Node id
n-e2303094356f15c9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5d4882089ebc7eac

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     2;
  2:    11,      4;
  3:    36,     16;
  4:   133,    104,    16;
  5:   454,    464,    96;
  6:  1551,   2060,   720,    64;
  7:  5192,   8416,  3968,   512;
  8: 17225,  33232, 20448,  4352,  256;
  9: 56586, 126688, 96192, 28160, 2560;
%F Formula
The row polynomials satisfy (2*n^3+2*n^2-7*n+(8*n^2-6*n)*x+8*n*x^2)*P(n,x) = (10*n^3+8*n^2-33*n+9+(8*n^3+40*n^2-50*n+6)*x+(32*n^2+16*n-32)*x^2+(32*n-32)*x^3)*P(n-1,x) - (12*n^2+36*n+18+(28*n^2+132*n+102)*x+(16*n^2+160*n+212)*x^2+(64*n+192)*x^3+64*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[((10*n^3 + 8*n^2 - 33*n + 9 + (8*n^3 + 40*n^2 - 50*n + 6)*x + (32*n^2 + 16*n - 32)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] - (12*n^2 + 36*n + 18 + (28*n^2 + 132*n + 102)*x + (16*n^2 + 160*n + 212)*x^2 + (64*n + 192)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n^2 - 7*n + (8*n^2 - 6*n)*x + 8*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] = ((10*n^3 + 8*n^2 - 33*n + 9 + (8*n^3 + 40*n^2 - 50*n + 6)*x + (32*n^2 + 16*n - 32)*x^2 + (32*n - 32)*x^3)*v[i - 1] - (12*n^2 + 36*n + 18 + (28*n^2 + 132*n + 102)*x + (16*n^2 + 160*n + 212)*x^2 + (64*n + 192)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n^2 - 7*n + (8*n^2 - 6*n)*x + 8*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

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