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	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

All 15 rows

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.