Real-rooted atlas

A207607 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(8 - 16\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(8 + 42\,x + 44\,x^{2} + 30\,x^{3} - 16\,n - 78\,n x - 82\,n x^{2} - 45\,n x^{3} + 8\,n^{2} + 36\,n^{2} x + 38\,n^{2} x^{2} + 15\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 30\,x^{2} + 80\,x^{3} + 60\,x^{4} - 4\,n x - 30\,n x^{2} - 56\,n x^{3} - 30\,n x^{4}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 18\,x^{4} + 15\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
229819
3215332171
42217611155265
5227137324355144989
6233216714125110983773691
7239313133532564527331698713775
82454282241704513581156469837258451409
9251561348613455333815313352248287796765191861
10257712512423485715051462311980661698778326517711716035
11263881720938296138474347291603378710942540648238727463682672279
12269106897955921124823873922815737462374845247540816907946792841213939973081
1327512731293687715418662144564636529566748792899656584060765211018192051831781137220045
142811496166861254556720122642421774165616968432276691953302092627953838158627785400081832040138907099
Technical details
Node id
n-ba332ed9c8481855 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2b83e2879aa49b31

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: 2,  3;
  2: 2,  9,   8;
  3: 2, 15,  33,   21;
  4: 2, 21,  76,  111,    55;
  5: 2, 27, 137,  324,   355,   144;
  6: 2, 33, 216,  714,  1251,  1098,   377;
  7: 2, 39, 313, 1335,  3256,  4527,  3316,   987;
  8: 2, 45, 428, 2241,  7045, 13581, 15646,  9837,  2584;
  9: 2, 51, 561, 3486, 13455, 33381, 53133, 52248, 28779, 6765;
%F Formula
The row polynomials satisfy (8*n^2-16*n+8)*P(n,x) = (8*n^2-16*n+8+(36*n^2-78*n+42)*x+(38*n^2-82*n+44)*x^2+(15*n^2-45*n+30)*x^3)*P(n-1,x) - ((4*n-4)*x+(30*n-30)*x^2+(56*n-80)*x^3+(30*n-60)*x^4)*d/dx P(n-1,x) + (3*x^3+18*x^4+15*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 8 + (36*n^2 - 78*n + 42)*x + (38*n^2 - 82*n + 44)*x^2 + (15*n^2 - 45*n + 30)*x^3)*P[n - 1, x] - ((4*n - 4)*x + (30*n - 30)*x^2 + (56*n - 80)*x^3 + (30*n - 60)*x^4)*D[P[n - 1, x], x] + (3*x^3 + 18*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 8 + (36*n^2 - 78*n + 42)*x + (38*n^2 - 82*n + 44)*x^2 + (15*n^2 - 45*n + 30)*x^3)*v[i - 1] - ((4*n - 4)*x + (30*n - 30)*x^2 + (56*n - 80)*x^3 + (30*n - 60)*x^4)*deriv(v[i - 1], x) + (3*x^3 + 18*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8)); 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