A207606 → reverse

nonnegative coefficients

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

Recurrence

\(\left(18 + 24\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(18 + 45\,x + 18\,x^{2} + 24\,n + 66\,n x + 30\,n x^{2} + 8\,n^{2} + 24\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x - 33\,x^{2} - 18\,x^{3} - 4\,n x - 22\,n x^{2} - 24\,n x^{3}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 12\,x^{4}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 2 + 7\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	2	7	4								13
1	2	11	16	5							34
2	2	15	36	30	6						89
3	2	19	64	91	50	7					233
4	2	23	100	204	196	77	8				610
5	2	27	144	385	540	378	112	9			1597
6	2	31	196	650	1210	1254	672	156	10		4181
7	2	35	256	1015	2366	3289	2640	1122	210	11	10946

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-5561b91e0c2d41c3 · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-c538a4320031e08b

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 ?

A048575: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A122367: 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,  7,   4;
  1: 2, 11,  16,    5;
  2: 2, 15,  36,   30,    6;
  3: 2, 19,  64,   91,   50,     7;
  4: 2, 23, 100,  204,  196,    77,     8;
  5: 2, 27, 144,  385,  540,   378,   112,     9;
  6: 2, 31, 196,  650, 1210,  1254,   672,   156,   10;
  7: 2, 35, 256, 1015, 2366,  3289,  2640,  1122,  210,   11;
  8: 2, 39, 324, 1496, 4200,  7371,  8008,  5148, 1782,  275,  12;
  9: 2, 43, 400, 2109, 6936, 14756, 20384, 17875, 9438, 2717, 352, 13;

%F Formula

The row polynomials satisfy (8*n^2+24*n+18)*P(n,x) = (8*n^2+24*n+18+(24*n^2+66*n+45)*x+(12*n^2+30*n+18)*x^2)*P(n-1,x) - ((4*n+6)*x+(22*n+33)*x^2+(24*n+18)*x^3)*d/dx P(n-1,x) + (3*x^3+12*x^4)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2 + 7*x + 4*x^2.

%t Mathematica

P[0, x_] := 2 + 7*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 24*n + 18 + (24*n^2 + 66*n + 45)*x + (12*n^2 + 30*n + 18)*x^2)*P[n - 1, x] - ((4*n + 6)*x + (22*n + 33)*x^2 + (24*n + 18)*x^3)*D[P[n - 1, x], x] + (3*x^3 + 12*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 7*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 24*n + 18 + (24*n^2 + 66*n + 45)*x + (12*n^2 + 30*n + 18)*x^2)*v[i - 1] - ((4*n + 6)*x + (22*n + 33)*x^2 + (24*n + 18)*x^3)*deriv(v[i - 1], x) + (3*x^3 + 12*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18)); 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-6ecf89a2d2d3d1a7
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-6ecf89a2d2d3d1a7
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-6ecf89a2d2d3d1a7

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 reversal, shift 0 e-6ecf89a2d2d3d1a7

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the 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-62ac8b7225a515c7 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3222de37fcca30b3 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A207606: Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023