A145597 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(18 + 2\,x + 9\,n + 9\,n x + n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(5\,x - 5\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	3							3
2	15	12						27
3	45	90	30					165
4	105	378	315	60				858
5	210	1176	1764	840	105			4095
6	378	3024	7056	6048	1890	168		18564
7	630	6804	22680	30240	17010	3780	252	81396

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f884ff92b7eaa34d · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-590a65e5a7de1331

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:    3;
  2:   15,    12;
  3:   45,    90,     30;
  4:  105,   378,    315,     60;
  5:  210,  1176,   1764,    840,    105;
  6:  378,  3024,   7056,   6048,   1890,    168;
  7:  630,  6804,  22680,  30240,  17010,   3780,   252;
  8:  990, 13860,  62370, 118800, 103950,  41580,  6930,   360;
  9: 1485, 26136, 152460, 392040, 490050, 304920, 91476, 11880, 495;

%F Formula

The row polynomials satisfy (n^2+5*n-6)*P(n,x) = (n^2+9*n+18+(3*n^2+9*n+2)*x)*P(n-1,x) + ((2*n+5)*x-(2*n+5)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 9*n + 18 + (3*n^2 + 9*n + 2)*x)*P[n - 1, x] + ((2*n + 5)*x - (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n - 6)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 9*n + 18 + (3*n^2 + 9*n + 2)*x)*v[i - 1] + ((2*n + 5)*x - (2*n + 5)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n - 6)); 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-dd8d38abcb6c1917
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-dd8d38abcb6c1917
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-dd8d38abcb6c1917
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-dd8d38abcb6c1917

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-dd8d38abcb6c1917

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-bcb2e3135d28c083 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-e8333902099cfe55
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-bcb2e3135d28c083 (expanded above)

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