A115951 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 26\,x^{2} + 24\,x^{3} + 8\,x^{4} - 2\,n - 10\,n x - 22\,n x^{2} - 22\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 20\,x^{2} + 36\,x^{3} + 28\,x^{4} + 8\,x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	2	8						10
3	0	0	12	32					44
4	0	0	6	72	136				214
5	0	0	0	60	400	592			1052
6	0	0	0	20	480	2160	2624		5284
7	0	0	0	0	280	3360	11424	11776	26840

All 15 rows

Technical details

Node id: n-8ba32770adecde3b · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-e6a1708012e30eb7

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: 0, 2;
  2: 0, 2,  8;
  3: 0, 0, 12, 32;
  4: 0, 0,  6, 72, 136;
  5: 0, 0,  0, 60, 400,  592;
  6: 0, 0,  0, 20, 480, 2160,  2624;
  7: 0, 0,  0,  0, 280, 3360, 11424,  11776;
  8: 0, 0,  0,  0,  70, 2800, 21840,  59584,  53344;
  9: 0, 0,  0,  0,   0, 1260, 23520, 135072, 307584, 243392;

%F Formula

The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(2*n-2+(10*n-12)*x+(22*n-26)*x^2+(22*n-24)*x^3+(8*n-8)*x^4)*P(n-1,x) + (4*x+20*x^2+36*x^3+28*x^4+8*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[(-(2*n - 2 + (10*n - 12)*x + (22*n - 26)*x^2 + (22*n - 24)*x^3 + (8*n - 8)*x^4)*P[n - 1, x] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*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] = (-(2*n - 2 + (10*n - 12)*x + (22*n - 26)*x^2 + (22*n - 24)*x^3 + (8*n - 8)*x^4)*v[i - 1] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A008556 → x ↦ x+1

References

[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