Real-rooted atlas

A086645 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-6 - 4\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(6\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3924841
42716214432365
58186414407681283281
624340501080010080384051229525
772917496680409676862208184322048265721
82187714423810247620487257603548168601681922391485
96561279936195955252254726842880486604819169283932163276821523361
1019683106288294478403233260855427328533744643019161699532801769472131072193710245
11590493936600433026001847577604003084804981616643773952001769472005013504078643205242881743392201
1217714714289858190531440990763488264203596841098337283985293312248168448099267379224654643234603008209715215690529805
135314415101833681062467250438868481621249344030743691264369545379842923875532815431565312537919488011890851841509949448388608141214768241
1415943231796270583353038416245889483849367218432021232361779230883435315230091552358420061034905691782512640283398635525643436032654311424335544321270932914165
154782969624974616135411166801155508623365144764584961371937222656236970974822427776817561602268440100864130472371814452646746521614586111590426424115200281857228813421772811438396227481
Technical details
Node id
n-995a901d38d5f8be · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-38d9a649e263b473

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:     3,       2;
   3:     9,      24,       8;
   4:    27,     162,     144,       32;
   5:    81,     864,    1440,      768,      128;
   6:   243,    4050,   10800,    10080,     3840,      512;
   7:   729,   17496,   68040,    96768,    62208,    18432,     2048;
   8:  2187,   71442,  381024,   762048,   725760,   354816,    86016,    8192;
   9:  6561,  279936, 1959552,  5225472,  6842880,  4866048,  1916928,  393216,   32768;
  10: 19683, 1062882, 9447840, 32332608, 55427328, 53374464, 30191616, 9953280, 1769472, 131072;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(2*n-4)*x)*P(n-1,x) + (6*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (2*n - 4)*x)*P[n - 1, x] + (6*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = ((3*n - 6 + (2*n - 4)*x)*v[i - 1] + (6*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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. [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