A102662 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	6							10
2	9	22	12						43
3	22	76	80	24					202
4	53	236	368	232	48				937
5	128	698	1440	1384	608	96			4354
6	309	1994	5116	6616	4496	1504	192		20227
7	746	5560	17056	27728	25600	13312	3584	384	93970

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-3dddab772a6867d2 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-5104a571b27debd0

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,     6;
  2:    9,    22,     12;
  3:   22,    76,     80,     24;
  4:   53,   236,    368,    232,     48;
  5:  128,   698,   1440,   1384,    608,     96;
  6:  309,  1994,   5116,   6616,   4496,   1504,    192;
  7:  746,  5560,  17056,  27728,  25600,  13312,   3584,   384;
  8: 1801, 15224,  54336, 106416, 124384,  88320,  36992,  8320,   768;
  9: 4348, 41102, 167296, 383344, 542656, 489920, 280832, 98176, 18944, 1536;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (2 + 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] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + (2 + 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-30bb9f6b8a15c298
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-30bb9f6b8a15c298
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-30bb9f6b8a15c298
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-30bb9f6b8a15c298

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 A013609 e-30bb9f6b8a15c298

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