Real-rooted atlas

A099093 → 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

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(12 + 56\,x + 96\,x^{2} + 72\,x^{3} + 20\,x^{4} - 9\,n - 44\,n x - 81\,n x^{2} - 66\,n x^{3} - 20\,n x^{4}\right)\,P_{n-1}(x) + \left(7 + 45\,x + 113\,x^{2} + 139\,x^{3} + 84\,x^{4} + 20\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
212454299
3452524682881053
4171126935194320198011259
5648599422113406623726013608120285
6245727216125307306909421686308124935281285227
79315120204663390202953637169284074624247536064281613732173
835316520263334700112280734281070814108644037457640194711044418064146723643
913389322171321629047769704550191402595349755246425290824331813584150718320303652801567691901
1050762793333877710948937694330112073649402647588761402519731441892364082856276432115199496020870006416750252683
1119245603890235635695776619625175727183046223183770723163353320976443640482752396944699042403208569687156673921434392064178970728365
12729656116082323216226621469910114938408014855641193002048352540101436193968007244564513494470763645609677281984712297766538745088098585527681912241103579
13276633636602771707265467692487999044362232186897337342632662761786923217341325744580511044478997010284492968968376326338208856016139264281924871038965126775766899220431643042781
1410487977226949504333211724098823529176298011837869598254326676611435118468904374352468617141432539337689656790477029373849004333251915727228591841042928129537603535683606948066240465697330560218305127249451
Technical details
Node id
n-7ff0d8a4bed93ba2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a49d109cdb28249d

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:      3,       6;
  2:     12,      45,       42;
  3:     45,     252,      468,      288;
  4:    171,    1269,     3519,     4320,      1980;
  5:    648,    5994,    22113,    40662,     37260,     13608;
  6:   2457,   27216,   125307,   306909,    421686,    308124,     93528;
  7:   9315,  120204,   663390,  2029536,   3716928,   4074624,   2475360,    642816;
  8:  35316,  520263,  3347001, 12280734,  28107081,  41086440,  37457640,  19471104,   4418064;
  9: 133893, 2217132, 16290477, 69704550, 191402595, 349755246, 425290824, 331813584, 150718320, 30365280;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(9*n-12+(44*n-56)*x+(81*n-96)*x^2+(66*n-72)*x^3+(20*n-20)*x^4)*P(n-1,x) + (7+45*x+113*x^2+139*x^3+84*x^4+20*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[(-(9*n - 12 + (44*n - 56)*x + (81*n - 96)*x^2 + (66*n - 72)*x^3 + (20*n - 20)*x^4)*P[n - 1, x] + (7 + 45*x + 113*x^2 + 139*x^3 + 84*x^4 + 20*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] = (-(9*n - 12 + (44*n - 56)*x + (81*n - 96)*x^2 + (66*n - 72)*x^3 + (20*n - 20)*x^4)*v[i - 1] + (7 + 45*x + 113*x^2 + 139*x^3 + 84*x^4 + 20*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

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741