A064861 → 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

\(P_{n}(x) = \left(6 + 28\,x + 32\,x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 6 + 28\,x + 32\,x^{2}\)

\(P_{1}(x) = 18 + 132\,x + 320\,x^{2} + 256\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	28	32								66
1	18	132	320	256							726
2	36	336	1168	1792	1024						4356
3	108	1296	6192	14720	17408	8192					47916
4	216	3024	17568	54208	93696	86016	32768				287496
5	648	10800	76896	303168	714752	1007616	786432	262144			3162456
6	1296	24192	196992	913920	2642176	4874240	5603328	3670016	1048576		18974736
7	3888	82944	784512	4317696	15237888	35760128	55803904	55836672	32505856	8388608	208722096

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2826700b30d60082 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-35d669a8896afbf8

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:     6,     28,      32;
  1:    18,    132,     320,      256;
  2:    36,    336,    1168,     1792,      1024;
  3:   108,   1296,    6192,    14720,     17408,      8192;
  4:   216,   3024,   17568,    54208,     93696,     86016,      32768;
  5:   648,  10800,   76896,   303168,    714752,   1007616,     786432,     262144;
  6:  1296,  24192,  196992,   913920,   2642176,   4874240,    5603328,    3670016,    1048576;
  7:  3888,  82944,  784512,  4317696,  15237888,  35760128,   55803904,   55836672,   32505856,    8388608;
  8:  7776, 181440, 1900800, 11773440,  47746560, 132471808,  254648320,  334888960,  288358400,  146800640,   33554432;
  9: 23328, 606528, 7153920, 50526720, 237427200, 779387904, 1823719424, 3041853440, 3544186880, 2747269120, 1275068416, 268435456;

%F Formula

The row polynomials satisfy P(n,x) = (6+28*x+32*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 28*x + 32*x^2, P(1,x) = 18 + 132*x + 320*x^2 + 256*x^3.

%t Mathematica

P[0, x_] := 6 + 28*x + 32*x^2;
P[1, x_] := 18 + 132*x + 320*x^2 + 256*x^3;
P[n_, x_] := P[n, x] = Expand[(6 + 28*x + 32*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 28*x + 32*x^2; v[2] = 18 + 132*x + 320*x^2 + 256*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 28*x + 32*x^2)*v[i - 2]); 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-5e9e3e2d2cafb589
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-5e9e3e2d2cafb589
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-5e9e3e2d2cafb589

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-5e9e3e2d2cafb589

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.