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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	1801	15224	54336	106416	124384	88320	36992	8320	768							436561
9	4348	41102	167296	383344	542656	489920	280832	98176	18944	1536						2028154
10	10497	109726	501580	1316368	2189216	2402240	1755136	840320	251648	42496	3072					9422299
11	25342	290244	1472112	4353832	8320512	10758144	9575424	5850752	2399232	627712	94208	6144				43773658
12	61181	761892	4245744	13971416	30170640	44937984	47226752	35202944	18432256	6599680	1531904	206848	12288			203361529
13	147704	1987074	12067872	43732376	105312288	177616416	215421184	189860992	121371136	55490048	17612800	3672064	450560	24576		944767090
14	356589	5153810	33879420	134063400	356202800	671136672	923177920	940220800	711302400	397186560	160936960	45840384	8671232	974848	49152	4389152947

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