A232535 → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 20\,n + n x + 22\,n^{2} - 3\,n^{2} x - 8\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-12\,x - 15\,n x - n x^{2} + 6\,n^{2} + 53\,n^{2} x - 7\,n^{2} x^{2} - 8\,n^{3} - 22\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2} + 8\,n x - 8\,n x^{2} - 16\,n^{2} x + 20\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	8	6						14
3	18	50	12					80
4	32	196	168	20				416
5	50	540	1008	420	30			2048
6	72	1210	3960	3564	880	42		9728
7	98	2366	12012	18876	10010	1638	56	45056

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-299240b586a1358b · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-e452e6c864432806

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): no

OEIS matches ?

No OEIS matches.

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   2;
  2:   8,    6;
  3:  18,   50,    12;
  4:  32,  196,   168,     20;
  5:  50,  540,  1008,    420,     30;
  6:  72, 1210,  3960,   3564,    880,     42;
  7:  98, 2366, 12012,  18876,  10010,   1638,    56;
  8: 128, 4200, 30576,  74360,  68640,  24024,  2800,   72;
  9: 162, 6936, 68544, 238680, 340340, 206856, 51408, 4488, 90;

%F Formula

The row polynomials satisfy -(8*n^3-22*n^2+20*n-6-(2*n^3-3*n^2+n)*x)*P(n,x) = -(8*n^3-6*n^2+(22*n^3-53*n^2+15*n+12)*x-(6*n^3-7*n^2-n)*x^2)*P(n-1,x) - ((16*n^2-8*n-4)*x-(20*n^2-8*n-4)*x^2+4*n^2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 6*n^2 + (22*n^3 - 53*n^2 + 15*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*P[n - 1, x] - ((16*n^2 - 8*n - 4)*x - (20*n^2 - 8*n - 4)*x^2 + 4*n^2*x^3)*D[P[n - 1, x], x])/(-8*n^3 + 22*n^2 - 20*n + 6 + (2*n^3 - 3*n^2 + n)*x)]];
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; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 6*n^2 + (22*n^3 - 53*n^2 + 15*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*v[i - 1] - ((16*n^2 - 8*n - 4)*x - (20*n^2 - 8*n - 4)*x^2 + 4*n^2*x^3)*deriv(v[i - 1], x))/(-8*n^3 + 22*n^2 - 20*n + 6 + (2*n^3 - 3*n^2 + n)*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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, 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-16137bf2f4ea840e
Coefficients of one sign. Every nonzero row, n >= 0: 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-16137bf2f4ea840e
Hadamard kernel (Malo). Rows P_n, n >= 0, 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-16137bf2f4ea840e
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-16137bf2f4ea840e

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-16137bf2f4ea840e

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a75bd3d9d36c6f35 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023