Real-rooted atlas

A290595 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-5 - 13\,x - 11\,x^{2} - 3\,x^{3} + 3\,n + 9\,n x + 9\,n x^{2} + 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
34272758
4283066754051414
528043141539019845850548334
63640732243700627597806889052296352125246
758240145677696790682787598839803400277858357577955114237262
811065603327211227693932410418625002070185040225823059012806743952955402457257810766

All 15 rows

Technical details
Node id
n-76d87447f0c0edac · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6c53b1c3395f48c0

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:         1,           3;
   3:         4,          27,           27;
   4:        28,         306,          675,           405;
   5:       280,        4314,        15390,         19845,          8505;
   6:      3640,       73224,       370062,        759780,        688905,         229635;
   7:     58240,     1456776,      9679068,      27875988,      39803400,       27785835,        7577955;
   8:   1106560,    33272112,    276939324,    1041862500,    2070185040,     2258230590,     1280674395,     295540245;
   9:  24344320,   858360240,   8652648096,   40605917220,  104767753860,   158081193270,   139297968810,   66496555125,   13299311025;
  10: 608608000, 24691383360, 293982898176, 1666138572648, 5338219912740, 10409190131340, 12629222776170, 9327250132200, 3843500886225, 678264862275;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3*n-5+(9*n-13)*x+(9*n-11)*x^2+(3*n-3)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*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[((3*n - 5 + (9*n - 13)*x + (9*n - 11)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((3*n - 5 + (9*n - 13)*x + (9*n - 11)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + 2*x + 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

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