A102662 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 4\,x\right)\,P_{n-2}(x) + \left(2 + 4\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	12							16
2	9	44	48						101
3	22	152	320	192					686
4	53	472	1472	1856	768				4621
5	128	1396	5760	11072	9728	3072			31156
6	309	3988	20464	52928	71936	48128	12288		210041
7	746	11120	68224	221824	409600	425984	229376	49152	1416026

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-baf4dd53b457b011 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-3b4658f27559a394

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:    4,    12;
  2:    9,    44,     48;
  3:   22,   152,    320,     192;
  4:   53,   472,   1472,    1856,     768;
  5:  128,  1396,   5760,   11072,    9728,     3072;
  6:  309,  3988,  20464,   52928,   71936,    48128,    12288;
  7:  746, 11120,  68224,  221824,  409600,   425984,   229376,    49152;
  8: 1801, 30448, 217344,  851328, 1990144,  2826240,  2367488,  1064960,  196608;
  9: 4348, 82204, 669184, 3066752, 8682496, 15677440, 17973248, 12566528, 4849664, 786432;

%F Formula

The row polynomials satisfy P(n,x) = (1+4*x)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 2, x] + (2 + 4*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] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 2] + (2 + 4*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-38bf0b24f025cc0b
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-38bf0b24f025cc0b
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-38bf0b24f025cc0b
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-38bf0b24f025cc0b

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 A013611 e-38bf0b24f025cc0b

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
  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-5b6052467d550b8d root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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