A084534 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-2 - 13\,x - 25\,x^{2} - 19\,x^{3} - 5\,x^{4} + 2\,n + 13\,n x + 25\,n x^{2} + 19\,n x^{3} + 5\,n x^{4}\right)\,P_{n-1}(x) + \left(-x - 8\,x^{2} - 18\,x^{3} - 16\,x^{4} - 5\,x^{5}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	3							4
2	1	6	7						14
3	1	9	24	18					52
4	1	12	50	84	47				194
5	1	15	85	225	275	123			724
6	1	18	129	468	900	864	322		2702
7	1	21	182	840	2219	3339	2639	843	10084

All 15 rows

Technical details

Node id: n-5c75d50b93f7f35c · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-800b3c5f1e1db21a

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,  3;
  2: 1,  6,   7;
  3: 1,  9,  24,   18;
  4: 1, 12,  50,   84,   47;
  5: 1, 15,  85,  225,  275,   123;
  6: 1, 18, 129,  468,  900,   864,   322;
  7: 1, 21, 182,  840, 2219,  3339,  2639,   843;
  8: 1, 24, 244, 1368, 4610,  9528, 11756,  7896,  2207;
  9: 1, 27, 315, 2079, 8532, 22518, 38121, 39825, 23256, 5778;

%F Formula

The row polynomials satisfy (2*n-2+(4*n-4)*x+(2*n-2)*x^2)*P(n,x) = (2*n-2+(13*n-13)*x+(25*n-25)*x^2+(19*n-19)*x^3+(5*n-5)*x^4)*P(n-1,x) - (x+8*x^2+18*x^3+16*x^4+5*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (13*n - 13)*x + (25*n - 25)*x^2 + (19*n - 19)*x^3 + (5*n - 5)*x^4)*P[n - 1, x] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*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] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (13*n - 13)*x + (25*n - 25)*x^2 + (19*n - 19)*x^3 + (5*n - 5)*x^4)*v[i - 1] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*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

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 read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

A086645 → basis A013611 → x ↦ x/(x+1)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023