Real-rooted atlas

A091042 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-12 - 71\,x - 138\,x^{2} - 111\,x^{3} - 32\,x^{4} + 8\,n + 46\,n x + 84\,n x^{2} + 62\,n x^{3} + 16\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 28\,x^{2} - 60\,x^{3} - 52\,x^{4} - 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
241317
31692121229
464528136810933053
525627521060817224984140681
61024135686937617022419929288573542057
74096645124108801349888240393621921967971617222669
81638429900822794249386240224902403127612823316976717445396238853
96553613598721207091259758592180125440337857920384393760242137808645700811282339921
102621446094848617349123570524161297572864306820992047137420804531003680246980562058113073317086609217
111048576270008323072983042033319936865217740824710740992479159385606222071424051729715440248433388605230176601227671469749
12419430411848908814968422401115140915254386294784182151266304427093008384700428353280786702325440575891476720247125844424470715894133033621093533
1316777216515899392716282265659331051520326293585920125416813363234520150999046850246828032971984938752096106027331206281850705936243595475215242364430472140421652081721
146710886422313697283377253580830788603084818850132787208179245711360258602650828806028177691443210353802875494412933704962816011415310922598467379302777488238299921405883812798742493538600539302297
15268435456959656755215725284556815646263869441055444801945651033136168960182296875171840488442245283840985901748264960149107626460979216625042886978561324552878771712712576339983808231627523606484343151886824417176612681496669
Technical details
Node id
n-ddfd53fb7e5f6d25 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3c37411ec544ec34

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 -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:      1;
   2:      4,      13;
   3:     16,      92,      121;
   4:     64,     528,     1368,      1093;
   5:    256,    2752,    10608,     17224,       9841;
   6:   1024,   13568,    69376,    170224,     199292,      88573;
   7:   4096,   64512,   410880,   1349888,    2403936,    2192196,     797161;
   8:  16384,  299008,  2279424,   9386240,   22490240,   31276128,   23316976,    7174453;
   9:  65536, 1359872, 12070912,  59758592,  180125440,  337857920,  384393760,  242137808,   64570081;
  10: 262144, 6094848, 61734912, 357052416, 1297572864, 3068209920, 4713742080, 4531003680, 2469805620, 581130733;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (8*n-12+(46*n-71)*x+(84*n-138)*x^2+(62*n-111)*x^3+(16*n-32)*x^4)*P(n-1,x) - (4*x+28*x^2+60*x^3+52*x^4+16*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (46*n - 71)*x + (84*n - 138)*x^2 + (62*n - 111)*x^3 + (16*n - 32)*x^4)*P[n - 1, x] - (4*x + 28*x^2 + 60*x^3 + 52*x^4 + 16*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (46*n - 71)*x + (84*n - 138)*x^2 + (62*n - 111)*x^3 + (16*n - 32)*x^4)*v[i - 1] - (4*x + 28*x^2 + 60*x^3 + 52*x^4 + 16*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)); 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

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