Real-rooted atlas

A091044 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 - 2\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
2605612128
3448608256321344
43344595237449608014080
5249605497646080180483200192147456
61863044890885125762711047481698564481544192
71390592423833653329923558400134144027955228672102416171008
810379520360222725290905642657792204467205879808967680798722304169345024
97747379230161817650656051247904768027904000010274201623543808315801621504051201773404160
10578272256249571532847176243205120630784351382528015803719684666224648785920098334725632001126418571329536
11431628288020452548608429747732485266039603241618341888221387980808022196224195741286430980505629466624144179224576194481487872
12322171740161662764810243844590714885249758822404696974745602886691389441241679953923734218342476993945601042595840855162883620864532482036636581888
13240472260608134274008678433879938826245101393379328509782769664035559994982401772347981824634847035392161976582144287123865603373793280241532928894566411468821327935176704
141794909388800107809664204802947799457792048522440540160535717618483204182714689126423717966856192987077895782430089937715206624711475201023594823681055843942466645196821790720245760223349036810240

Other recurrences

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

\(P_{n}(x) = \left(-4 - 8\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 4\,x\)
Technical details
Node id
n-7ca1770372a339f7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-9ad1bee5a2efcea7)
Fingerprint
f6-0878106a64e78b92

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:        8,         4;
  2:       60,        56,        12;
  3:      448,       608,       256,        32;
  4:     3344,      5952,      3744,       960,        80;
  5:    24960,     54976,     46080,     18048,      3200,       192;
  6:   186304,    489088,    512576,    271104,     74816,      9856,      448;
  7:  1390592,   4238336,   5332992,   3558400,   1341440,    279552,    28672,    1024;
  8: 10379520,  36022272,  52909056,  42657792,  20446720,   5879808,   967680,   79872,   2304;
  9: 77473792, 301618176, 506560512, 479047680, 279040000, 102742016, 23543808, 3158016, 215040, 5120;
%F Formula
The row polynomials satisfy n*P(n,x) = (10*n-2+(6*n-2)*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((10*n - 2 + (6*n - 2)*x)*P[n - 1, x] - (6 + 10*x + 4*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((10*n - 2 + (6*n - 2)*x)*v[i - 1] - (6 + 10*x + 4*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 - 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