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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	7776	90720	475200	1471680	2984160	4139744	3978880	2616320	1126400	286720	32768							17210368
9	23328	303264	1788480	6315840	14839200	24355872	28495616	23764480	13844480	5365760	1245184	131072						120472576
10	46656	653184	4183488	16208640	42310080	78390144	105702976	104520192	75217920	38420480	13221888	2752512	262144					481890304
11	139968	2146176	15163200	65359872	191764800	404410752	630669504	736372480	643734528	416133120	193347584	61145088	11796480	1048576				3373232128
12	279936	4572288	34618752	161046144	514249344	1192351104	2070160512	2734083968	2760214016	2119735296	1218961408	508985344	145883136	25690112	2097152			13492928512
13	839808	14836608	122145408	621613440	2186932608	5634050688	10979885952	16482893952	19216977920	17400061952	12135825408	6402801664	2473590784	660602880	109051904	8388608		94450499584
14	1679616	31352832	273964032	1487517696	5617092096	15641966592	33227873280	54925559808	71399743744	73234079744	59071774720	37077254144	17752784896	6268387328	1539309568	234881024	16777216	377801998336

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.