A064861 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 14\,x + 8\,x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 6 + 14\,x + 8\,x^{2}\)

\(P_{1}(x) = 18 + 66\,x + 80\,x^{2} + 32\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	14	8								28
1	18	66	80	32							196
2	36	168	292	224	64						784
3	108	648	1548	1840	1088	256					5488
4	216	1512	4392	6776	5856	2688	512				21952
5	648	5400	19224	37896	44672	31488	12288	2048			153664
6	1296	12096	49248	114240	165136	152320	87552	28672	4096		614656
7	3888	41472	196128	539712	952368	1117504	871936	436224	126976	16384	4302592

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-990897f38932c9f7 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-409ecd303d9e1d72

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,     14,       8;
  1:    18,     66,      80,      32;
  2:    36,    168,     292,     224,       64;
  3:   108,    648,    1548,    1840,     1088,      256;
  4:   216,   1512,    4392,    6776,     5856,     2688,      512;
  5:   648,   5400,   19224,   37896,    44672,    31488,    12288,     2048;
  6:  1296,  12096,   49248,  114240,   165136,   152320,    87552,    28672,     4096;
  7:  3888,  41472,  196128,  539712,   952368,  1117504,   871936,   436224,   126976,   16384;
  8:  7776,  90720,  475200, 1471680,  2984160,  4139744,  3978880,  2616320,  1126400,  286720,   32768;
  9: 23328, 303264, 1788480, 6315840, 14839200, 24355872, 28495616, 23764480, 13844480, 5365760, 1245184, 131072;

%F Formula

The row polynomials satisfy P(n,x) = (6+14*x+8*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 14*x + 8*x^2, P(1,x) = 18 + 66*x + 80*x^2 + 32*x^3.

%t Mathematica

P[0, x_] := 6 + 14*x + 8*x^2;
P[1, x_] := 18 + 66*x + 80*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[(6 + 14*x + 8*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 + 14*x + 8*x^2; v[2] = 18 + 66*x + 80*x^2 + 32*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 14*x + 8*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-72b81ad6731299bb
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-72b81ad6731299bb
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-72b81ad6731299bb

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-72b81ad6731299bb

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.