A199881 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-2}(x) + \left(x + x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 2\,x\)

\(P_{1}(x) = x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	1	2															3
1	0	1	2														3
2	0	1	4	4													9
3	0	0	2	7	6												15
4	0	0	1	7	15	10											33
5	0	0	0	3	16	28	16										63
6	0	0	0	1	11	38	53	26									129
7	0	0	0	0	4	30	82	97	42								255
8	0	0	0	0	1	16	79	173	176	68							513
9	0	0	0	0	0	5	50	191	352	315	110						1023
10	0	0	0	0	0	1	22	145	443	701	559	178					2049
11	0	0	0	0	0	0	6	77	386	986	1368	984	288				4095
12	0	0	0	0	0	0	1	29	244	974	2130	2628	1721	466			8193
13	0	0	0	0	0	0	0	7	112	707	2346	4484	4980	2993	754		16383
14	0	0	0	0	0	0	0	1	37	385	1925	5450	9242	9329	5180	1220	32769

Technical details

Node id: n-22333aafcaf54c94 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-92bdcb9f4d02545f

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1, 2;
  1: 0, 1, 2;
  2: 0, 1, 4, 4;
  3: 0, 0, 2, 7,  6;
  4: 0, 0, 1, 7, 15, 10;
  5: 0, 0, 0, 3, 16, 28, 16;
  6: 0, 0, 0, 1, 11, 38, 53,  26;
  7: 0, 0, 0, 0,  4, 30, 82,  97,  42;
  8: 0, 0, 0, 0,  1, 16, 79, 173, 176,  68;
  9: 0, 0, 0, 0,  0,  5, 50, 191, 352, 315, 110;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (x+2*x^2+x^3)*P(n-2,x) + (x+x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1 + 2*x, P(1,x) = x + 2*x^2.

%t Mathematica

P[0, x_] := 1 + 2*x;
P[1, x_] := x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((x + 2*x^2 + x^3)*P[n - 2, x] + (x + x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = ((x + 2*x^2 + x^3)*v[i - 2] + (x + x^2)*v[i - 1])/(1 + x)); 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 Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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