A391533 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

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

\(P_{0}(x) = 9 + 24\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	9	24	16								49
1	18	84	128	64							294
2	36	240	592	640	256						1764
3	72	624	2144	3648	3072	1024					10584
4	144	1536	6784	15872	20736	14336	4096				63504
5	288	3648	19712	58880	104960	111616	65536	16384			381024
6	576	8448	54016	196608	445440	643072	577536	294912	65536		2286144
7	1152	19200	141824	609280	1677312	3067904	3727360	2899968	1310720	262144	13716864

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d021e887e4fd1eea · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-19b2ced286a4e317

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): yes

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:    9,    24,     16;
  1:   18,    84,    128,      64;
  2:   36,   240,    592,     640,      256;
  3:   72,   624,   2144,    3648,     3072,     1024;
  4:  144,  1536,   6784,   15872,    20736,    14336,     4096;
  5:  288,  3648,  19712,   58880,   104960,   111616,    65536,     16384;
  6:  576,  8448,  54016,  196608,   445440,   643072,   577536,    294912,     65536;
  7: 1152, 19200, 141824,  609280,  1677312,  3067904,  3727360,   2899968,   1310720,   262144;
  8: 2304, 43008, 360448, 1785856,  5791744, 12845056, 19726336,  20709376,  14221312,  5767168,  1048576;
  9: 4608, 95232, 892928, 5013504, 18726912, 48857088, 90832896, 120324096, 111280128, 68419584, 25165824, 4194304;

%F Formula

The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 16*x^2.

%t Mathematica

P[0, x_] := 9 + 24*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(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] = 9 + 24*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (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-3d28f25fd9699ebd
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-3d28f25fd9699ebd
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-3d28f25fd9699ebd

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-3d28f25fd9699ebd

Geometric basis map (elementary) polytool ?
- premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check
    
    Parent-row provers of polytool polytool ?

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