Real-rooted atlas

Hadamard-factor parent of A155495 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
188117
2324818199
3128256160321577
4512128011204005013363
5204861446912358484072119601
6819228672394242688094081568981114243
732768131072212992180224844802150426881281665857
81310725898241105920111820865894422809644352432016213880899
952428826214405570560655360046592002050048549120844806600200122619537
10209715211534336273940483676569630638080164003845637632120806415100896802421131836323
1183886085033164813212057619922944019051315212032409650692096140574722471040256256137282881768398401
1233554432218103808627048448104962457611331174408255569924127784961412136963236147247590404164161892833814478554083
1313421772893952409629360128005402263552649959833653692334083111714816127008768036118118469701632871270465228825480392126102926097
145368709124026531840135895449602726297600036175872000334265057282205220864010478223360357212160085995520014189260815275520990080336004501152139002499
1521474836481717986918462277025792135291469824196293427200200655503360148562247680806480773123213321830493139763201926299648275185664257986561462272435205121886731088897

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 8 + 8\,x + x^{2}\)
\(P_{2}(x) = 32 + 48\,x + 18\,x^{2} + x^{3}\)
Technical details
Node id
n-2d5c09b3ab5c188a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-6e1e59d586c4c010)
Fingerprint
f6-607690c148a1ebbd

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:       8,        8,        1;
   2:      32,       48,       18,        1;
   3:     128,      256,      160,       32,        1;
   4:     512,     1280,     1120,      400,       50,        1;
   5:    2048,     6144,     6912,     3584,      840,       72,       1;
   6:    8192,    28672,    39424,    26880,     9408,     1568,      98,       1;
   7:   32768,   131072,   212992,   180224,    84480,    21504,    2688,     128,      1;
   8:  131072,   589824,  1105920,  1118208,   658944,   228096,   44352,    4320,    162,    1;
   9:  524288,  2621440,  5570560,  6553600,  4659200,  2050048,  549120,   84480,   6600,  200,   1;
  10: 2097152, 11534336, 27394048, 36765696, 30638080, 16400384, 5637632, 1208064, 151008, 9680, 242, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n+3*n*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 8 + 8*x + x^2.
%t Mathematica
P[1, x_] := 8 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 3*n*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 8 + 8*x + x^2; for(n = 2, N, my(i = n); v[i] = ((4*n + 3*n*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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