Real-rooted atlas

A082985 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-9 - 44\,x - 79\,x^{2} - 62\,x^{3} - 18\,x^{4} + 13\,n + 59\,n x + 97\,n x^{2} + 69\,n x^{3} + 18\,n x^{4}\right)\,P_{n-1}(x) + \left(-5 - 34\,x - 90\,x^{2} - 116\,x^{3} - 73\,x^{4} - 18\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14711
211373179
329143234127533
476493119412795113553
51991600512581726484204723627
65214998199054211449909314018191157039
713641520372384190795300625283077147454327671043669
83571453472511867925791557865195286315243166772031310716936001
993491332378416413092524728314911399157118550017898298305834452428746094987
10244763868252744250115070663157914259258577769935606838398039729375136314852097151306335887
116407911122228754890412504121292529472827695924406893404892048103790364851951904756011835483886072035832213
12167761317233227436326143494320505426495126297010222955715443057668602296189397020346689459407668462628430473355443113529634721
134392048987263847105444869457301904476930535125975311112785646172678229882007541183517245321030106386190694462739448114085068413421772789914587851
14114985125314433258271249161846095869590103612171751054750725497771900755620081221904958911260001030329721544385154416727478208882552894921316689536870911597549990319
Technical details
Node id
n-2e1dfc62107def93 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-fabb9dc8a447cb10

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,      7;
  2:   11,     37,     31;
  3:   29,    143,    234,     127;
  4:   76,    493,   1194,    1279,     511;
  5:  199,   1600,   5125,    8172,    6484,     2047;
  6:  521,   4998,  19905,   42114,   49909,    31401,     8191;
  7: 1364,  15203,  72384,  190795,  300625,   283077,   147454,   32767;
  8: 3571,  45347, 251186,  792579, 1557865,  1952863,  1524316,  677203,  131071;
  9: 9349, 133237, 841641, 3092524, 7283149, 11399157, 11855001, 7898298, 3058344, 524287;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (13*n-9+(59*n-44)*x+(97*n-79)*x^2+(69*n-62)*x^3+(18*n-18)*x^4)*P(n-1,x) - (5+34*x+90*x^2+116*x^3+73*x^4+18*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[((13*n - 9 + (59*n - 44)*x + (97*n - 79)*x^2 + (69*n - 62)*x^3 + (18*n - 18)*x^4)*P[n - 1, x] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((13*n - 9 + (59*n - 44)*x + (97*n - 79)*x^2 + (69*n - 62)*x^3 + (18*n - 18)*x^4)*v[i - 1] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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

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