Real-rooted atlas

A113953 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(8 + 40\,x + 72\,x^{2} + 56\,x^{3} + 16\,x^{4} - 7\,n - 34\,n x - 63\,n x^{2} - 52\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}(x) + \left(3 + 25\,x + 73\,x^{2} + 99\,x^{3} + 64\,x^{4} + 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
23182445
3548144128325
41113458810887042525
521324193255207488384019125
643786583222376465604953620992146125
785181616248786562216963624963174401146881113125

All 15 rows

Technical details
Node id
n-ff1d3304ea3a0ff3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e590fa3c506d7049

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:   3,   18,     24;
  3:   5,   48,    144,    128;
  4:  11,  134,    588,   1088,     704;
  5:  21,  324,   1932,   5520,    7488,    3840;
  6:  43,  786,   5832,  22376,   46560,   49536,    20992;
  7:  85, 1816,  16248,  78656,  221696,  362496,   317440,   114688;
  8: 171, 4158,  43380, 253008,  899856, 1992960,  2677248,  1990656,   626688;
  9: 341, 9324, 111348, 760848, 3271824, 9163584, 16682496, 19003392, 12275712, 3424256;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(7*n-8+(34*n-40)*x+(63*n-72)*x^2+(52*n-56)*x^3+(16*n-16)*x^4)*P(n-1,x) + (3+25*x+73*x^2+99*x^3+64*x^4+16*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[(-(7*n - 8 + (34*n - 40)*x + (63*n - 72)*x^2 + (52*n - 56)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] + (3 + 25*x + 73*x^2 + 99*x^3 + 64*x^4 + 16*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] = (-(7*n - 8 + (34*n - 40)*x + (63*n - 72)*x^2 + (52*n - 56)*x^3 + (16*n - 16)*x^4)*v[i - 1] + (3 + 25*x + 73*x^2 + 99*x^3 + 64*x^4 + 16*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