A299989 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 6\,x + x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	x²⁹	P_n(1)
0	1	1																													2
1	8	14	7	1																											30
2	64	160	148	64	13	1																									450
3	512	1664	2208	1560	636	150	19	1																							6750
4	4096	16384	28160	27392	16656	6576	1688	272	25	1																					101250
5	32768	155648	327680	404480	325760	179936	69616	18880	3520	430	31	1																			1518750
6	262144	1441792	3588096	5357568	5360640	3798528	1962304	748672	211056	43440	6348	624	37	1																	22781250
7	2097152	13107200	37617664	65830912	78618624	67909632	43850240	21561728	8142784	2362528	522480	86520	10388	854	43	1															341718750
8	16777216	117440512	381681664	765460480	1061552128	1080819712	836878336	503504896	238362880	89318656	26497792	6189568	1124704	155680	15856	1120	49	1													5125781250
9	134217728	1040187392	3774873600	8531214336	13466861568	15781330944	14241497088	10130128896	5764810752	2648231424	986257152	297821952	72632832	14183232	2185632	259776	22968	1422	55	1											76886718750
10	1073741824	9126805504	36574330880	91939143680	162697052160	215583031296	222086823936	182271344640	121140756480	65904844800	29544256512	10948349952	3354251520	847084800	175217280	29375232	3928032	408960	31940	1760	61	1									1153300781250
11	8589934592	79456894976	348429221888	964085940224	1889785610240	2792785707008	3232889831424	3006274732032	2284840943616	1436354641920	752923877376	330757183488	122068368384	37850537472	9838498560	2133390336	382892928	56215104	6637312	614680	42988	2134	67	1							17299511718750
12	68719476736	687194767360	3272765079552	9882719748096	21251229745152	34645085257728	44509618503680	46240322551808	39549265772544	28206157529088	16926359814144	8599955374080	3714013925376	1365971693568	427879581696	113948651520	25701984000	4880468736	773282048	100956416	10669296	889680	56328	2544	73	1					259492675781250
13	549755813888	5909874999296	30374008717312	99385543229440	232578921529344	414550780280832	585198689320960	671625376694272	638345679994880	509185177419776	344197089460224	198563959406592	98238203461632	41811812474880	15332880740352	4844838395904	1317187362816	307204305408	61171052800	10327812352	1464374912	172089632	16458000	1248000	72176	2990	79	1			3892390136718750
14	4398046511104	50577534877696	279001075548160	983238273138688	2487318640328704	4811265314652160	7401473117782016	9298745929760768	9721716389445632	8575180876021760	6447033460195328	4162879389433856	2321486473592832	1122487679975424	471772124233728	172567804084224	54939410018304	15205597016064	3649781617664	756853121024	134852926208	20490778880	2628576704	280821632	24523408	1704976	90748	3472	85	1	58385852050781250

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Technical details

Node id: n-1f565ad1ec7a715f · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-cfb916af429c76b6

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,          1;
  1:         8,         14,          7,          1;
  2:        64,        160,        148,         64,          13,           1;
  3:       512,       1664,       2208,       1560,         636,         150,          19,           1;
  4:      4096,      16384,      28160,      27392,       16656,        6576,        1688,         272,         25,          1;
  5:     32768,     155648,     327680,     404480,      325760,      179936,       69616,       18880,       3520,        430,        31,         1;
  6:    262144,    1441792,    3588096,    5357568,     5360640,     3798528,     1962304,      748672,     211056,      43440,      6348,       624,       37,        1;
  7:   2097152,   13107200,   37617664,   65830912,    78618624,    67909632,    43850240,    21561728,    8142784,    2362528,    522480,     86520,    10388,      854,      43,      1;
  8:  16777216,  117440512,  381681664,  765460480,  1061552128,  1080819712,   836878336,   503504896,  238362880,   89318656,  26497792,   6189568,  1124704,   155680,   15856,   1120,    49,    1;
  9: 134217728, 1040187392, 3774873600, 8531214336, 13466861568, 15781330944, 14241497088, 10130128896, 5764810752, 2648231424, 986257152, 297821952, 72632832, 14183232, 2185632, 259776, 22968, 1422, 55, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(8 + 6*x + 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 + x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 6*x + 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-298e29fac11cdc8c
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-298e29fac11cdc8c
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-298e29fac11cdc8c

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 A007318 e-298e29fac11cdc8c

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.