Real-rooted atlas

A091042 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2} - 4\,x^{3} + 2\,n + 8\,n x + 10\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
35201641
475611264239
591204325762561393
61122012322816281610248119
713364291299841664013312409647321
81556060482880070400921606144016384275807
9178161142471808239360452608487424278528655361607521
101911402006416051269555217704962723840249036812451842621449369319
1121154033264329472179379258705921204224015597568123863045505024104857654608393
122320245262463148842099201714585644843008768737288591769660293120241172484194304318281039
13252600800801144000915200045260800146227200317521920466944000458752000288358400104857600167772161855077841
14273276117936197683218670080109983744428654592114307891221180579842724986880238760755213589544964529848326710886410812186007
152940601688963281408360954882493870081151016960368325427283414876161346345369615386804224121634816006325010432194615705626843545663018038201
Technical details
Node id
n-b134200100920bae · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3f225bab20c1f22f

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:  3,    4;
   3:  5,   20,    16;
   4:  7,   56,   112,     64;
   5:  9,  120,   432,    576,    256;
   6: 11,  220,  1232,   2816,   2816,    1024;
   7: 13,  364,  2912,   9984,  16640,   13312,    4096;
   8: 15,  560,  6048,  28800,  70400,   92160,   61440,   16384;
   9: 17,  816, 11424,  71808, 239360,  452608,  487424,  278528,   65536;
  10: 19, 1140, 20064, 160512, 695552, 1770496, 2723840, 2490368, 1245184, 262144;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (2*n-1+(8*n-6)*x+(10*n-9)*x^2+(4*n-4)*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*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[((2*n - 1 + (8*n - 6)*x + (10*n - 9)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*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] = ((2*n - 1 + (8*n - 6)*x + (10*n - 9)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*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

Candidate equalities ?

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