Real-rooted atlas

A122076 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1358
27221746
3188412965296
4472926746842571954
5123955294545053415102512968
6322300011580236922708116386409786158
78439163424831088711664881518587646116385572552

All 15 rows

Technical details
Node id
n-4d01a7d8730237d2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c277d1d0c4fa4cc9

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:    2;
  1:    3,     5;
  2:    7,    22,     17;
  3:   18,    84,    129,      65;
  4:   47,   292,    674,     684,     257;
  5:  123,   955,   2945,    4505,    3415,    1025;
  6:  322,  3000,  11580,   23692,   27081,   16386,    4097;
  7:  843,  9163,  42483,  108871,  166488,  151858,   76461,   16385;
  8: 2207, 27416, 148396,  457016,  875650, 1068536,  810708,  349528,   65537;
  9: 5778, 80748, 499761, 1797573, 4140126, 6330564, 6424959, 4172463, 1572867, 262145;
%F Formula
The row polynomials satisfy (2*n-2+(4*n-4)*x+(2*n-2)*x^2)*P(n,x) = (13*n-13+(59*n-59)*x+(97*n-97)*x^2+(69*n-69)*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 >= 2, with P(0,x) = 2, P(1,x) = 3 + 5*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n - 13 + (59*n - 59)*x + (97*n - 97)*x^2 + (69*n - 69)*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 - 2 + (4*n - 4)*x + (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = ((13*n - 13 + (59*n - 59)*x + (97*n - 97)*x^2 + (69*n - 69)*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 - 2 + (4*n - 4)*x + (2*n - 2)*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