A207606 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 34 + 63\,x + 23\,x^{2} + 2\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	13	13	2								28
1	34	63	23	2							122
2	89	273	176	35	2						575
3	233	1118	1130	384	49	2					2916
4	610	4432	6592	3399	725	65	2				15825
5	1597	17220	36262	26602	8425	1243	83	2			91434
6	4181	66045	191938	192475	84852	18304	1988	103	2		559888
7	10946	251141	989473	1320186	778014	230041	36113	3016	125	2	3619057

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6fc37854ec05afc6 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-649cce3c34834eb6

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 ?

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:    13,      13,        2;
  1:    34,      63,       23,        2;
  2:    89,     273,      176,       35,        2;
  3:   233,    1118,     1130,      384,       49,        2;
  4:   610,    4432,     6592,     3399,      725,       65,       2;
  5:  1597,   17220,    36262,    26602,     8425,     1243,      83,       2;
  6:  4181,   66045,   191938,   192475,    84852,    18304,    1988,     103,      2;
  7: 10946,  251141,   989473,  1320186,   778014,   230041,   36113,    3016,    125,    2;
  8: 28657,  949465,  5006568,  8718114,  6681432,  2600038,  553070,   66170,   4389,  149,   2;
  9: 75025, 3575376, 24992832, 55995066, 54710124, 27251695, 7541555, 1211754, 114324, 6175, 175, 2;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 13 + 13*x + 2*x^2, P(1,x) = 34 + 63*x + 23*x^2 + 2*x^3.

%t Mathematica

P[0, x_] := 13 + 13*x + 2*x^2;
P[1, x_] := 34 + 63*x + 23*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 13 + 13*x + 2*x^2; v[2] = 34 + 63*x + 23*x^2 + 2*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + x)*v[i - 1] + x*deriv(v[i - 1], 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-d84114ea7bbb5d21
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-d84114ea7bbb5d21
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-d84114ea7bbb5d21

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 A008277 e-d84114ea7bbb5d21

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- 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.
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023