A308737 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 - n + n x\right)\,P_{n-1}(x) + \left(1 - x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	1														4
2	7	8	1													16
3	15	31	17	1												64
4	31	94	96	34	1											256
5	63	253	382	258	67	1										1024
6	127	636	1275	1280	645	132	1									4096
7	255	1531	3831	5115	3845	1545	261	1								16384
8	511	3578	10738	17906	17920	10766	3598	518	1							65536
9	1023	8185	28652	57316	71666	57358	28700	8212	1031	1						262144
10	2047	18424	73701	171984	258006	258048	172074	73776	18459	2056	1					1048576
11	4095	40951	184285	491445	860070	1032150	860202	491610	184395	40995	4105	1				4194304
12	8191	90102	450516	1351570	2703195	3784572	3784704	2703492	1351845	450670	90156	8202	1			16777216
13	16383	196597	1081290	3604326	8109805	12975831	15138684	12976260	8110377	3604755	1081498	196662	16395	1		67108864
14	32767	425972	2555839	9371440	23428691	42171844	56229459	56229888	42172845	23429692	9372077	2556112	426049	32780	1	268435456

Technical details

Node id: n-a0baa26a40b2e44f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-6fbba7c3cf69e7d1

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    3,    1;
  2:    7,    8,     1;
  3:   15,   31,    17,     1;
  4:   31,   94,    96,    34,     1;
  5:   63,  253,   382,   258,    67,     1;
  6:  127,  636,  1275,  1280,   645,   132,     1;
  7:  255, 1531,  3831,  5115,  3845,  1545,   261,    1;
  8:  511, 3578, 10738, 17906, 17920, 10766,  3598,  518,    1;
  9: 1023, 8185, 28652, 57316, 71666, 57358, 28700, 8212, 1031, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n-4-n*x)*P(n-1,x) + (1-x^2)*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[-(n - 4 - n*x)*P[n - 1, x] + (1 - x^2)*D[P[n - 1, x], x]];
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] = -(n - 4 - n*x)*v[i - 1] + (1 - x^2)*deriv(v[i - 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 (−∞, 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 of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - 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 ?

A308737: Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,…

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