Real-rooted atlas

A207607 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(24 - 48\,n + 24\,n^{2}\right)\,P_{n}(x) = \left(72 + 126\,x + 44\,x^{2} + 10\,x^{3} - 144\,n - 234\,n x - 82\,n x^{2} - 15\,n x^{3} + 72\,n^{2} + 108\,n^{2} x + 38\,n^{2} x^{2} + 5\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(36\,x + 90\,x^{2} + 80\,x^{3} + 20\,x^{4} - 36\,n x - 90\,n x^{2} - 56\,n x^{3} - 10\,n x^{4}\right)\,P_{n-1}'(x) + \left(9\,x^{3} + 18\,x^{4} + 5\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
21827853
3541359921309
4162567684333551801
5486218736992916106514410497
614588019174961927811259329437761181
74374284317605910813587912407439948987356589

All 15 rows

Technical details
Node id
n-a385d417b7856bd2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4f553049ae0af005

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:     6,      3;
  2:    18,     27,       8;
  3:    54,    135,      99,      21;
  4:   162,    567,     684,     333,      55;
  5:   486,   2187,    3699,    2916,    1065,     144;
  6:  1458,   8019,   17496,   19278,   11259,    3294,     377;
  7:  4374,  28431,   76059,  108135,   87912,   40743,    9948,    987;
  8: 13122,  98415,  312012,  544563,  570645,  366687,  140814,  29511,  2584;
  9: 39366, 334611, 1226907, 2541294, 3269565, 2703861, 1434591, 470232, 86337, 6765;
%F Formula
The row polynomials satisfy (24*n^2-48*n+24)*P(n,x) = (72*n^2-144*n+72+(108*n^2-234*n+126)*x+(38*n^2-82*n+44)*x^2+(5*n^2-15*n+10)*x^3)*P(n-1,x) - ((36*n-36)*x+(90*n-90)*x^2+(56*n-80)*x^3+(10*n-20)*x^4)*d/dx P(n-1,x) + (9*x^3+18*x^4+5*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((72*n^2 - 144*n + 72 + (108*n^2 - 234*n + 126)*x + (38*n^2 - 82*n + 44)*x^2 + (5*n^2 - 15*n + 10)*x^3)*P[n - 1, x] - ((36*n - 36)*x + (90*n - 90)*x^2 + (56*n - 80)*x^3 + (10*n - 20)*x^4)*D[P[n - 1, x], x] + (9*x^3 + 18*x^4 + 5*x^5)*D[P[n - 1, x], {x, 2}])/(24*n^2 - 48*n + 24)]];
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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((72*n^2 - 144*n + 72 + (108*n^2 - 234*n + 126)*x + (38*n^2 - 82*n + 44)*x^2 + (5*n^2 - 15*n + 10)*x^3)*v[i - 1] - ((36*n - 36)*x + (90*n - 90)*x^2 + (56*n - 80)*x^3 + (10*n - 20)*x^4)*deriv(v[i - 1], x) + (9*x^3 + 18*x^4 + 5*x^5)*deriv(deriv(v[i - 1], x), x))/(24*n^2 - 48*n + 24)); 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

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023