Real-rooted atlas

A091044 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-4\,x - 14\,x^{2} - 16\,x^{3} - 6\,x^{4} + 4\,n + 20\,n x + 34\,n x^{2} + 24\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 12\,x^{2} - 24\,x^{3} - 20\,x^{4} - 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2166460140
3643847364481632
4256204859527424334419024
510241024039936757766982424960221760
64096491522406406144008616966287361863042585024
71638422937613516804341760820224091054085495808139059230133248
86553610485767225344279838726658048099581952913797124701388810379520351258880
92621444718592372244481688207364847861769136046081129365504882671616395645952774737924094573568
101048576209715201863843849688842243260809216742077235211559976960121672826888278437888328700723257827225647729847296
114194304922746889122611205347737600206454128645509428019210366694195213749452800012593455104075844845568270265630724316282880556379873280
1216777216402653184438304768028605153280124618014720381664886784842378772480134963055820815574545203201262186659840681751265280220329607168322171740166485639090176
136710886417448304642073663897614911589580872366424064025022127144966340252532736119171752919041661219477913616958404689920123207601356806031949168640178339930112024047226060875602149588992
14268435456751619276896838090752760746082304407005193830415683744169984448802397880329686406188236815841666636185619533265475993617872045998080011764513112064052662185492480143477650227201794909388800881283238707200
Technical details
Node id
n-2d0ce1c0452a8c5a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c761722d499074dd

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:      4,       8;
  2:     16,      64,       60;
  3:     64,     384,      736,       448;
  4:    256,    2048,     5952,      7424,      3344;
  5:   1024,   10240,    39936,     75776,     69824,     24960;
  6:   4096,   49152,   240640,    614400,    861696,    628736,     186304;
  7:  16384,  229376,  1351680,   4341760,   8202240,   9105408,    5495808,   1390592;
  8:  65536, 1048576,  7225344,  27983872,  66580480,  99581952,   91379712,  47013888,  10379520;
  9: 262144, 4718592, 37224448, 168820736, 484786176, 913604608, 1129365504, 882671616, 395645952, 77473792;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n+(20*n-4)*x+(34*n-14)*x^2+(24*n-16)*x^3+(6*n-6)*x^4)*P(n-1,x) - (2*x+12*x^2+24*x^3+20*x^4+6*x^5)*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[((4*n + (20*n - 4)*x + (34*n - 14)*x^2 + (24*n - 16)*x^3 + (6*n - 6)*x^4)*P[n - 1, x] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
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] = ((4*n + (20*n - 4)*x + (34*n - 14)*x^2 + (24*n - 16)*x^3 + (6*n - 6)*x^4)*v[i - 1] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^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

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