A207607 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,x + 13\,x^{2} + 4\,x^{3} - n + 8\,n x - 10\,n x^{2} - 4\,n x^{3} - 3\,n^{2} - 12\,n^{2} x - 2\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-2 + 20\,x + 40\,x^{2} + 17\,x^{3} + 2\,x^{4} - 7\,n - 5\,n x - 19\,n x^{2} - 11\,n x^{3} - 2\,n x^{4} - 7\,n^{2} - 12\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3} + 3\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 33\,x^{2} - 44\,x^{3} - 17\,x^{4} - 2\,x^{5} - 8\,n x - 10\,n x^{2} - 2\,n x^{3} - 4\,n^{2} x + 7\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	6	9	2						17
3	10	27	15	2					54
4	15	64	64	21	2				166
5	21	130	202	117	27	2			499
6	28	237	525	464	186	33	2		1475
7	36	399	1190	1476	890	271	39	2	4303

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4cacc7eae80c0b65 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9f2dd18ab5ef93be

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:  1;
  1:  3,   2;
  2:  6,   9,    2;
  3: 10,  27,   15,    2;
  4: 15,  64,   64,   21,     2;
  5: 21, 130,  202,  117,    27,    2;
  6: 28, 237,  525,  464,   186,   33,    2;
  7: 36, 399, 1190, 1476,   890,  271,   39,   2;
  8: 45, 632, 2436, 4014,  3355, 1520,  372,  45,  2;
  9: 55, 954, 4608, 9702, 10659, 6630, 2394, 489, 51, 2;

%F Formula

The row polynomials satisfy -(2*n^3+3*n^2+n-(4*n^3-12*n^2+8*n+6)*x+(10*n-13)*x^2+(4*n-4)*x^3)*P(n,x) = -(2*n^3+7*n^2+7*n+2-(3*n^3-12*n^2-5*n+20)*x-(2*n^3-2*n^2-19*n+40)*x^2+(11*n-17)*x^3+(2*n-2)*x^4)*P(n-1,x) - ((4*n^2+8*n+4)*x-(7*n^2-10*n-33)*x^2-(2*n^2-2*n-44)*x^3+17*x^4+2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 7*n^2 + 7*n + 2 - (3*n^3 - 12*n^2 - 5*n + 20)*x - (2*n^3 - 2*n^2 - 19*n + 40)*x^2 + (11*n - 17)*x^3 + (2*n - 2)*x^4)*P[n - 1, x] - ((4*n^2 + 8*n + 4)*x - (7*n^2 - 10*n - 33)*x^2 - (2*n^2 - 2*n - 44)*x^3 + 17*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-2*n^3 - 3*n^2 - n + (4*n^3 - 12*n^2 + 8*n + 6)*x - (10*n - 13)*x^2 - (4*n - 4)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 7*n^2 + 7*n + 2 - (3*n^3 - 12*n^2 - 5*n + 20)*x - (2*n^3 - 2*n^2 - 19*n + 40)*x^2 + (11*n - 17)*x^3 + (2*n - 2)*x^4)*v[i - 1] - ((4*n^2 + 8*n + 4)*x - (7*n^2 - 10*n - 33)*x^2 - (2*n^2 - 2*n - 44)*x^3 + 17*x^4 + 2*x^5)*deriv(v[i - 1], x))/(-2*n^3 - 3*n^2 - n + (4*n^3 - 12*n^2 + 8*n + 6)*x - (10*n - 13)*x^2 - (4*n - 4)*x^3)); 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-1945cad536bc16ce
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-1945cad536bc16ce
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-1945cad536bc16ce
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-1945cad536bc16ce

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-1945cad536bc16ce

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8f881604bf5a9d75 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3cffc405672d7710 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