Real-rooted atlas

A063007 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(-54 - 45\,x - 8\,x^{2} + 36\,n + 27\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-18\,x - 18\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2639
3545413121
4540810396631809
55670113408370270032128401
66123615309015120073710177301683458649
767359620207882500470163296059346011377889897544041
8750578426270244390685683199581015581160451105271887248639125700129

All 15 rows

Technical details
Node id
n-c44e96619cc36843 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-7c3d3d10489865b7

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:        54,         54,         13;
   4:       540,        810,        396,         63;
   5:      5670,      11340,       8370,       2700,        321;
   6:     61236,     153090,     151200,      73710,      17730,       1683;
   7:    673596,    2020788,    2500470,    1632960,     593460,     113778,       8989;
   8:   7505784,   26270244,   39068568,   31995810,   15581160,    4511052,     718872,     48639;
   9:  84440070,  337760280,  586702116,  577945368,  353076570,  136964520,   32942700,   4491288,   265729;
  10: 956987460, 4306443570, 8556593760, 9851341500, 7241830596, 3524591070, 1135632960, 233566740, 27823230, 1462563;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = (36*n-54+(27*n-45)*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[((36*n - 54 + (27*n - 45)*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] = ((36*n - 54 + (27*n - 45)*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