Real-rooted atlas

A122848 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3 + 3\,x\right)\,P_{n}(x) = \left(8 + 33\,x + 33\,x^{2} + 8\,x^{3} - 5\,n - 18\,n x - 21\,n x^{2} - 8\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 14\,x + 30\,x^{2} + 26\,x^{3} + 8\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
22132035
3439120112275
41012455510486882425
526400237567409040454423125
67613741005037835767407886431936236875
72324858424201993255415208460487006722368002571875
87641806418242010240723481625730945692173766365440184140829440625
926206926479578052006082124051356043540952311361002297605919667214943232352953125
109496276700355500026459580126027930400330413856625700121636560010949587205637491201260636164414421875
1135696113832416187600135245220736132980273492750370620562321264846440015373121280120620403205496088576110198374457367421875
1214015248470887548222069875080042751179901817885163654988050579119005662600182603163600193498581760134315334912548247982089954734080772423515625
1356850421209344359134620364951787224795434950118758202992411878165127104573933178019409898324962598169288000243589496550415140531927045591852032009271415603210746208203125
142390480956053841747589480193282096281443139177487684827250103005687126442875984917879519092534985524309351102700643664744379296030762778452224172928188160005827402190848888502796288154146098046875
Technical details
Node id
n-7674f5b97866799a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7ac044cd69385489

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:    1,     4;
  2:    2,    13,     20;
  3:    4,    39,    120,     112;
  4:   10,   124,    555,    1048,      688;
  5:   26,   400,   2375,    6740,     9040,     4544;
  6:   76,  1374,  10050,   37835,    76740,    78864,    31936;
  7:  232,  4858,  42420,  199325,   541520,   846048,   700672,    236800;
  8:  764, 18064, 182420, 1024072,  3481625,  7309456,  9217376,   6365440,  1841408;
  9: 2620, 69264, 795780, 5200608, 21240513, 56043540, 95231136, 100229760, 59196672, 14943232;
%F Formula
The row polynomials satisfy (3+3*x)*P(n,x) = -(5*n-8+(18*n-33)*x+(21*n-33)*x^2+(8*n-8)*x^3)*P(n-1,x) + (2+14*x+30*x^2+26*x^3+8*x^4)*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[(-(5*n - 8 + (18*n - 33)*x + (21*n - 33)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] + (2 + 14*x + 30*x^2 + 26*x^3 + 8*x^4)*D[P[n - 1, x], x])/(3 + 3*x)]];
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] = (-(5*n - 8 + (18*n - 33)*x + (21*n - 33)*x^2 + (8*n - 8)*x^3)*v[i - 1] + (2 + 14*x + 30*x^2 + 26*x^3 + 8*x^4)*deriv(v[i - 1], x))/(3 + 3*x)); 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. [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