A249252 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

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

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

\(P_{1}(x) = 5 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	4							9
2	17	24	8						49
3	61	124	80	16					281
4	217	576	552	224	32				1601
5	773	2532	3216	1968	576	64			9129
6	2753	10728	16968	13888	6176	1408	128		52049
7	9805	44300	83856	85968	51392	17856	3328	256	296761

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-2 - 2\,n + 4\,n^{2}\right)\,P_{n-1}(x) + \left(30 + 42\,x + 12\,x^{2} - 17\,n - 20\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(17 + 37\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

Technical details

Node id: n-20086f90792ae22f · revision 2 · JSON
Definition source: derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-a397357ef38eb84b)
Fingerprint: f6-205a62c6b3f27a2b

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 ?

A123270: 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:      1;
  1:      5,      4;
  2:     17,     24,       8;
  3:     61,    124,      80,      16;
  4:    217,    576,     552,     224,      32;
  5:    773,   2532,    3216,    1968,     576,      64;
  6:   2753,  10728,   16968,   13888,    6176,    1408,    128;
  7:   9805,  44300,   83856,   85968,   51392,   17856,   3328,    256;
  8:  34921, 179472,  395560,  487328,  366240,  171520,  48768,   7680,   512;
  9: 124373, 716468, 1801936, 2592752, 2348096, 1385536, 531712, 127744, 17408, 1024;

%F Formula

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (3 + 2*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] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (3 + 2*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-6b1a6b6617e41154
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-6b1a6b6617e41154, e-8b34bf66d342d123
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-6b1a6b6617e41154, e-8b34bf66d342d123
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-6b1a6b6617e41154, e-8b34bf66d342d123
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8b34bf66d342d123

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A007318 e-6b1a6b6617e41154

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-fb471901eb718e65 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
definition_equality e-a397357ef38eb84b root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image affine, shift 0 e-8b34bf66d342d123

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ theorem_check liu_wang_derivative_lag e-fb471901eb718e65 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003