Real-rooted atlas

A104684 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(-9 + 3\,x + 8\,x^{2} + 9\,n + 3\,n x - 4\,n x^{2}\right)\,P_{n-1}(x) + \left(18\,x + 18\,x^{2} + 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2336
39241346
42713517163396
581648140411523213606
624328359180124207515168333876
77291166452245102600100305480248989324556
82187459272704597172551000755764127302547486393151896

All 15 rows

Technical details
Node id
n-68e54766ae0dd6bf · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3d3f3d7068653107

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:     3,      3;
   3:     9,     24,      13;
   4:    27,    135,     171,       63;
   5:    81,    648,    1404,     1152,      321;
   6:   243,   2835,    9180,    12420,     7515,     1683;
   7:   729,  11664,   52245,   102600,   100305,    48024,     8989;
   8:  2187,  45927,  270459,   717255,  1000755,   764127,   302547,    48639;
   9:  6561, 174960, 1306368,  4463424,  8289540,  8932896,  5589360,  1886208,   265729;
  10: 19683, 649539, 5983632, 25474176, 60194988, 85505868, 74907504, 39679632, 11665971, 1462563;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = (9*n-9+(3*n+3)*x-(4*n-8)*x^2)*P(n-1,x) + (18*x+18*x^2+4*x^3)*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[((9*n - 9 + (3*n + 3)*x - (4*n - 8)*x^2)*P[n - 1, x] + (18*x + 18*x^2 + 4*x^3)*D[P[n - 1, x], x])/(3*n - 3)]];
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] = ((9*n - 9 + (3*n + 3)*x - (4*n - 8)*x^2)*v[i - 1] + (18*x + 18*x^2 + 4*x^3)*deriv(v[i - 1], x))/(3*n - 3)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023