A207606 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 13 + 33\,x + 18\,x^{2}\)

\(P_{1}(x) = 34 + 132\,x + 153\,x^{2} + 54\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	13	33	18								64
1	34	132	153	54							373
2	89	465	837	621	162						2174
3	233	1530	3753	4320	2349	486					12671
4	610	4824	15012	23598	19845	8505	1458				73852
5	1597	14772	55755	111510	127980	84564	29889	4374			430441
6	4181	44283	196569	478197	698625	629127	341901	102789	13122		2508794
7	10946	130620	666801	1912788	3402486	3898692	2883195	1329696	347733	39366	14622323

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-98aa86c8bf356480 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-55426929a3969ca5

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 ?

A038725: 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:    13,      33,      18;
  1:    34,     132,     153,       54;
  2:    89,     465,     837,      621,      162;
  3:   233,    1530,    3753,     4320,     2349,       486;
  4:   610,    4824,   15012,    23598,    19845,      8505,      1458;
  5:  1597,   14772,   55755,   111510,   127980,     84564,     29889,     4374;
  6:  4181,   44283,  196569,   478197,   698625,    629127,    341901,   102789,    13122;
  7: 10946,  130620,  666801,  1912788,  3402486,   3898692,   2883195,  1329696,   347733,    39366;
  8: 28657,  380415, 2195694,  7260570, 15247197,  21274407,  20003760, 12535884,  5019165,  1161297,  118098;
  9: 75025, 1096596, 7061526, 26456004, 64120815, 105666120, 120951306, 96289236, 52317414, 18502020, 3838185, 354294;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 33*x + 18*x^2, P(1,x) = 34 + 132*x + 153*x^2 + 54*x^3.

%t Mathematica

P[0, x_] := 13 + 33*x + 18*x^2;
P[1, x_] := 34 + 132*x + 153*x^2 + 54*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 13 + 33*x + 18*x^2; v[2] = 34 + 132*x + 153*x^2 + 54*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + 3*x)*v[i - 1]); 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-77ee9774535f66a4
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-77ee9774535f66a4
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-77ee9774535f66a4
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-77ee9774535f66a4

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

basis_map A013610 e-77ee9774535f66a4

Geometric basis map (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 ?

No candidate equalities or related sequences.

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