Real-rooted atlas

A078812 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-132 - 54\,x + 54\,x^{2} + 240\,n + 54\,n x - 54\,n x^{2} - 162\,n^{2} + 54\,n^{3}\right)\,P_{n}(x) = \left(72\,n + 36\,n x - 81\,n x^{2} - 81\,n x^{3} + 81\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(20 + 27\,x - 171\,x^{2} - 243\,x^{3} - 81\,x^{4} - 45\,n - 162\,n x - 81\,n x^{2} + 45\,n^{2} + 162\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2639
32636971
49623716227522
532511881431648813673
61044511290727182243024325083
73239199804743955080324008748729167615
89804731312188623392552916001363233061821871101780

All 15 rows

Technical details
Node id
n-e39162f26463abae · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4b7612e63eed9bde

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     6,      3;
   3:    26,     36,       9;
   4:    96,    237,     162,      27;
   5:   325,   1188,    1431,     648,       81;
   6:  1044,   5112,    9072,    7182,     2430,      243;
   7:  3239,  19980,   47439,   55080,    32400,     8748,     729;
   8:  9804,  73131,  218862,  339255,   291600,   136323,   30618,    2187;
   9: 29134, 255276,  925011, 1805976,  2067930,  1408428,  546021,  104976,   6561;
  10: 85350, 859749, 3665088, 8664354, 12481938, 11303145, 6368544, 2108268, 354294, 19683;
%F Formula
The row polynomials satisfy (54*n^3-162*n^2+240*n-132+(54*n-54)*x-(54*n-54)*x^2)*P(n,x) = (81*n^3+72*n+(81*n^3+36*n)*x-81*n*x^2-81*n*x^3)*P(n-1,x) + (45*n^2-45*n+20+(162*n^2-162*n+27)*x+(81*n^2-81*n-171)*x^2-243*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 + 72*n + (81*n^3 + 36*n)*x - 81*n*x^2 - 81*n*x^3)*P[n - 1, x] + (45*n^2 - 45*n + 20 + (162*n^2 - 162*n + 27)*x + (81*n^2 - 81*n - 171)*x^2 - 243*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 - 162*n^2 + 240*n - 132 + (54*n - 54)*x - (54*n - 54)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 + 72*n + (81*n^3 + 36*n)*x - 81*n*x^2 - 81*n*x^3)*v[i - 1] + (45*n^2 - 45*n + 20 + (162*n^2 - 162*n + 27)*x + (81*n^2 - 81*n - 171)*x^2 - 243*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 - 162*n^2 + 240*n - 132 + (54*n - 54)*x - (54*n - 54)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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