Real-rooted atlas

A110441 → basis A007318 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-4\,x - 8\,x^{2} + 2\,n + 12\,n x + 8\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 8\,x^{2} - 8\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2181423
31124448105
411690208164479
51201525449085602185
612423011202904377619129967
7128324200070801430415200652845465
813243432481462040384666085977622288207391
91365604928269649486421398429760023099276096946025
10140702710445808196224564200107276812881768805122598084315343
111448609840731043696001295168313984051512805438016331987288704019684665
121481034132001110606478082681952792806416582240238976002249558412405504302854489792639
1315212241724816214010723685126976179166724570416083946880107783424915118084601081610340096409593865
141561430220482290641694528919432837128960112006944250907392410406464474850304367066880169572352353032961868384047
Technical details
Node id
n-3bb53a141e5d1684 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-010ab161ddea77e9

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: 1,  4;
  2: 1,  8,  14;
  3: 1, 12,  44,   48;
  4: 1, 16,  90,  208,   164;
  5: 1, 20, 152,  544,   908,   560;
  6: 1, 24, 230, 1120,  2904,  3776,   1912;
  7: 1, 28, 324, 2000,  7080, 14304,  15200,   6528;
  8: 1, 32, 434, 3248, 14620, 40384,  66608,  59776,  22288;
  9: 1, 36, 560, 4928, 26964, 94864, 213984, 297600, 230992, 76096;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (2*n+(12*n-4)*x+(8*n-8)*x^2)*P(n-1,x) - (x+8*x^2+8*x^3)*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[((2*n + (12*n - 4)*x + (8*n - 8)*x^2)*P[n - 1, x] - (x + 8*x^2 + 8*x^3)*D[P[n - 1, x], x])/(2*n)]];
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] = ((2*n + (12*n - 4)*x + (8*n - 8)*x^2)*v[i - 1] - (x + 8*x^2 + 8*x^3)*deriv(v[i - 1], x))/(2*n)); 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

Candidate equalities ?

Same fingerprint ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023