Real-rooted atlas

A086645 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 72\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
251217
341168144353
43652196388817288177
5328126256722887603220736198593
6295252952601107360188352013478402488324912337
72657213188664151463523574195242612480223948802985984122336033
8239148533480804192514896581732928976727808876137472355332096358318083054149297

All 15 rows

Technical details
Node id
n-3d116785edf34455 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-5105b4356f23af0f

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:         5,         12;
   3:        41,        168,         144;
   4:       365,       2196,        3888,         1728;
   5:      3281,      26256,       72288,        76032,        20736;
   6:     29525,     295260,     1107360,      1883520,      1347840,       248832;
   7:    265721,    3188664,    15146352,     35741952,     42612480,     22394880,      2985984;
   8:   2391485,   33480804,   192514896,    581732928,    976727808,    876137472,    355332096,     35831808;
   9:  21523361,  344373792,  2324523456,   8566304256,  18590860800,  23786348544,  16805117952,   5446434816,   429981696;
  10: 193710245, 3486784428, 27022579776, 117533700864, 313102444032, 522064963584, 531994853376, 305716985856, 81266540544, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -(9+72*x+144*x^2)*P(n-2,x) + (10+24*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 72*x + 144*x^2)*P[n - 2, x] + (10 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 12*x; for(n = 3, N, my(i = n); v[i] = -(9 + 72*x + 144*x^2)*v[i - 2] + (10 + 24*x)*v[i - 1]); 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

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