A110552 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	4	5	1													10
3	8	17	10	1												36
4	16	49	51	19	1											136
5	32	129	196	134	36	1										528
6	64	321	645	650	330	69	1									2080
7	128	769	1926	2575	1940	783	134	1								8256
8	256	1793	5383	8981	8995	5411	1813	263	1							32896
9	512	4097	14344	28700	35896	28742	14392	4124	520	1						131328
10	1024	9217	36873	86052	129108	129150	86142	36948	9252	1033	1					524800
11	2048	20481	92170	245805	430200	516306	430332	245970	92280	20525	2058	1				2098176
12	4096	45057	225291	675895	1351845	1892682	1892814	1352142	676170	225445	45111	4107	1			8390656
13	8192	98305	540684	1802306	4055260	6488559	7570200	6488988	4055832	1802735	540892	98370	8204	1		33558528
14	16384	212993	1277965	4685902	11714846	21086923	28116231	28116660	21087924	11715847	4686539	1278238	213070	16397	1	134225920

Technical details

Node id: n-ca2c0ee85b709a1a · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-ef4f1d5c279ca087

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:   2,    1;
  2:   4,    5,     1;
  3:   8,   17,    10,     1;
  4:  16,   49,    51,    19,     1;
  5:  32,  129,   196,   134,    36,     1;
  6:  64,  321,   645,   650,   330,    69,     1;
  7: 128,  769,  1926,  2575,  1940,   783,   134,    1;
  8: 256, 1793,  5383,  8981,  8995,  5411,  1813,  263,   1;
  9: 512, 4097, 14344, 28700, 35896, 28742, 14392, 4124, 520, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) - (x+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[(2 + n*x)*P[n - 1, x] - (x + 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] = (2 + n*x)*v[i - 1] - (x + 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 A triangular array related to A077028 and distributing the values of A007582.?
  - 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 ?

A110552: A triangular array related to A077028 and distributing the values of A007582.

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