A299989 → 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

\(P_{n}(x) = \left(8 + 12\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	1	2															3
1	8	28	28	8													72
2	64	320	592	512	208	32											1728
3	512	3328	8832	12480	10176	4800	1216	128									41472
4	4096	32768	112640	219136	266496	210432	108032	34816	6400	512							995328
5	32768	311296	1310720	3235840	5212160	5757952	4455424	2416640	901120	220160	31744	2048					23887872
6	262144	2883584	14352384	42860544	85770240	121552896	125587456	95830016	54030336	22241280	6500352	1277952	151552	8192			573308928
7	2097152	26214400	150470656	526647296	1257897984	2173108224	2806415360	2759901184	2084552704	1209614336	535019520	177192960	42549248	6995968	704512	32768	13759414272

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-bae5a2c240bb2e7e · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-80a60edfe80ff887

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
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:         1,          2;
  1:         8,         28,          28,           8;
  2:        64,        320,         592,         512,          208,           32;
  3:       512,       3328,        8832,       12480,        10176,         4800,         1216,           128;
  4:      4096,      32768,      112640,      219136,       266496,       210432,       108032,         34816,          6400,           512;
  5:     32768,     311296,     1310720,     3235840,      5212160,      5757952,      4455424,       2416640,        901120,        220160,         31744,         2048;
  6:    262144,    2883584,    14352384,    42860544,     85770240,    121552896,    125587456,      95830016,      54030336,      22241280,       6500352,      1277952,       151552,         8192;
  7:   2097152,   26214400,   150470656,   526647296,   1257897984,   2173108224,   2806415360,    2759901184,    2084552704,    1209614336,     535019520,    177192960,     42549248,      6995968,      704512,      32768;
  8:  16777216,  234881024,  1526726656,  6123683840,  16984834048,  34586230784,  53560213504,   64448626688,   61020897280,   45731151872,   27133739008,  12676235264,   4606787584,   1275330560,   259784704,   36700160,    3211264,    131072;
  9: 134217728, 2080374784, 15099494400, 68249714688, 215469785088, 505002590208, 911455813632, 1296656498688, 1475791552512, 1355894489088, 1009927323648, 609939357696, 297504079872, 116189036544, 35809394688, 8512339968, 1505230848, 186384384, 14417920, 524288;

%F Formula

The row polynomials satisfy P(n,x) = (8+12*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x + 4*x^2)*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 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 12*x + 4*x^2)*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-1cab7244184e4365
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-1cab7244184e4365
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-1cab7244184e4365

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-1cab7244184e4365

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-0e7ef206328a472c root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.