Real-rooted atlas

A125662 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(8 + 8\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(3 + 8\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2323264
3184384192760
49282976307210248000
54364189442995220480512078860
61964810771223040024064012288024576745856
7858725690881537920217088016896006881281146886856176
836729628556169335808166451201748992010838016367001652428861726080
91545492137917445293900811420057615149568012661555265142784188743682359296546964500
106419936646752962851706887224985601159495680122280345684777369637276876894371840104857604786463680
1126393128296268928147579552042967040008098979840103301775369032368128534773760020525875204613734404613734441464522984
1210758281613315454407395624960243352268805271994368078973009920834891939846230900736032180797440109576192002214592512201326592356215470784
1343540022058905753603609597696013248753664032445692928055861759180869346512076862555317862440706113536018639486976057025757184104689827848724152323038825469980
14175143616025713957760172331735040698126796800190754545664037128906178565297253384192561002997350444110530478082543782461440104602586316829051427225648855252992375809638425769632352000
Technical details
Node id
n-bf6741ae8e850c3e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d54e0f7c6bf98dc6

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       4;
  2:      32,       32;
  3:     184,      384,      192;
  4:     928,     2976,     3072,      1024;
  5:    4364,    18944,    29952,     20480,      5120;
  6:   19648,   107712,   230400,    240640,    122880,     24576;
  7:   85872,   569088,  1537920,   2170880,   1689600,    688128,   114688;
  8:  367296,  2855616,  9335808,  16645120,  17489920,  10838016,  3670016,   524288;
  9: 1545492, 13791744, 52939008, 114200576, 151495680, 126615552, 65142784, 18874368, 2359296;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (4*n+8+(4*n+8)*x)*P(n-1,x) + (3+8*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8 + (4*n + 8)*x)*P[n - 1, x] + (3 + 8*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8 + (4*n + 8)*x)*v[i - 1] + (3 + 8*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023