A110552 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 2\,x + 4\,n - 7\,n x + 6\,n^{2} x - n^{3} x\right)\,P_{n-1}(x) + \left(-3 - 4\,x - x^{2} + n - n x - 2\,n x^{2} + n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 5 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	5	8													13
3	10	34	24												68
4	19	102	147	64											332
5	36	268	588	516	160										1568
6	69	660	1950	2580	1605	384									7248
7	134	1566	5820	10300	9630	4614	896								32960
8	263	3626	16233	35980	44905	32298	12551	2048							147904
9	520	8248	43176	114968	179480	172200	100408	32776	4608						656384
10	1033	18504	110844	344568	645750	774648	602364	294984	82953	10240					2885888
11	2058	41050	276840	983880	2151660	3097836	3011400	1966440	829530	204810	22528				12588032
12	4107	90222	676335	2704680	6760710	11356884	13248774	10814760	6083055	2252910	495627	49152			54537216
13	8204	196740	1622676	7210940	20279160	38933928	52991400	51908472	36497340	18023060	5947524	1179660	106496		234905600
14	16397	426140	3834714	18746156	58579235	126527544	196816620	224929848	189782307	117148460	51544922	15335580	2768909	229376	1006686208

Technical details

Node id: n-dc291c1c7ee3c297 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-09c263d417b67197

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   0;
  1:   2;
  2:   5,    8;
  3:  10,   34,    24;
  4:  19,  102,   147,     64;
  5:  36,  268,   588,    516,    160;
  6:  69,  660,  1950,   2580,   1605,    384;
  7: 134, 1566,  5820,  10300,   9630,   4614,    896;
  8: 263, 3626, 16233,  35980,  44905,  32298,  12551,  2048;
  9: 520, 8248, 43176, 114968, 179480, 172200, 100408, 32776, 4608;

%F Formula

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (4*n-8-(n^3-6*n^2+7*n+2)*x)*P(n-1,x) + (n-3+(n^2-n-4)*x+(n^2-2*n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 5 + 8*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 - (n^3 - 6*n^2 + 7*n + 2)*x)*P[n - 1, x] + (n - 3 + (n^2 - n - 4)*x + (n^2 - 2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 5 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 - (n^3 - 6*n^2 + 7*n + 2)*x)*v[i - 1] + (n - 3 + (n^2 - n - 4)*x + (n^2 - 2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n + 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.