A098158 → reverse

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1								1
3	1	1							2
4	1	3							4
5	1	6	1						8
6	1	10	5						16
7	1	15	15	1					32
8	1	21	35	7					64
9	1	28	70	28	1				128
10	1	36	126	84	9				256
11	1	45	210	210	45	1			512
12	1	55	330	462	165	11			1024
13	1	66	495	924	495	66	1		2048
14	1	78	715	1716	1287	286	13		4096
15	1	91	1001	3003	3003	1001	91	1	8192

Technical details

Node id: n-eb1314dd9b4fafe6 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-4e15cddc85973b47

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1,  1;
   4: 1,  3;
   5: 1,  6,   1;
   6: 1, 10,   5;
   7: 1, 15,  15,  1;
   8: 1, 21,  35,  7;
   9: 1, 28,  70, 28, 1;
  10: 1, 36, 126, 84, 9;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x))/(n - 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

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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) with diagonals T(n,n-k) = binomial(n, 2*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

Related ?

Same fingerprint ?

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