A099097 → D

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(6 + 54\,x - 9\,n - 27\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(-16 - 108\,x - 162\,x^{2} + 12\,n + 45\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 54\,x^{2} - 81\,x^{3} + 4\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	3														3
3	1	18													19
4	0	12	81												93
5	0	2	81	324											407
6	0	0	27	432	1215										1674
7	0	0	3	216	2025	4374									6618
8	0	0	0	48	1350	8748	15309								25455
9	0	0	0	4	450	7290	35721	52488							95953
10	0	0	0	0	75	3240	35721	139968	177147						356151
11	0	0	0	0	5	810	19845	163296	531441	590490					1305887
12	0	0	0	0	0	108	6615	108864	708588	1968300	1948617				4741092
13	0	0	0	0	0	6	1323	45360	551124	2952450	7144929	6377292			17072484
14	0	0	0	0	0	0	147	12096	275562	2624400	11908215	25509168	20726199		61055787
15	0	0	0	0	0	0	7	2016	91854	1530900	11908215	46766808	89813529	66961566	217074895

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Technical details

Node id: n-4634927692f4f2c9 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-2e4970c7c29b11f6

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 -2)
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

OEIS matches ?

A074361: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 0;
   2: 3;
   3: 1, 18;
   4: 0, 12, 81;
   5: 0,  2, 81, 324;
   6: 0,  0, 27, 432, 1215;
   7: 0,  0,  3, 216, 2025, 4374;
   8: 0,  0,  0,  48, 1350, 8748, 15309;
   9: 0,  0,  0,   4,  450, 7290, 35721,  52488;
  10: 0,  0,  0,   0,   75, 3240, 35721, 139968, 177147;

%F Formula

The row polynomials satisfy (3*n^2-9*n+6-(27*n-54)*x)*P(n,x) = -(2*n^2-12*n+16-(45*n-108)*x+162*x^2)*P(n-1,x) + ((4*n-8)*x+(9*n-54)*x^2-81*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 12*n + 16 - (45*n - 108)*x + 162*x^2)*P[n - 1, x] + ((4*n - 8)*x + (9*n - 54)*x^2 - 81*x^3)*D[P[n - 1, x], x])/(3*n^2 - 9*n + 6 - (27*n - 54)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 12*n + 16 - (45*n - 108)*x + 162*x^2)*v[i - 1] + ((4*n - 8)*x + (9*n - 54)*x^2 - 81*x^3)*deriv(v[i - 1], x))/(3*n^2 - 9*n + 6 - (27*n - 54)*x)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ab48d0fc2b7fbb7d
Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-ab48d0fc2b7fbb7d
Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-ab48d0fc2b7fbb7d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-ab48d0fc2b7fbb7d

Laguerre’s theorem on differential operators [SP14] polytool ?
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Related sequences ?

No candidate equalities or related sequences.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.