A152842 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 30\,x + 27\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 4 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	9														13
2	8	30	27													65
3	32	192	378	243												845
4	64	480	1332	1620	729											4225
5	256	2496	9648	18468	17496	6561										54925
6	512	5760	26784	65880	90396	65610	19683									274625
7	2048	27648	158976	504576	954504	1076004	669222	177147								3570125
8	4096	61440	400896	1486080	3422736	5015520	4566456	2361960	531441							17850625
9	16384	282624	2156544	9552384	27065664	50866704	63405504	50545944	23383404	4782969						232058125
10	32768	614400	5160960	25574400	82788480	182930400	279411120	291308400	198404640	79716150	14348907					1160290625
11	131072	2752512	26173440	148746240	561323520	1476817920	2764018080	3679933680	3415394160	2104506360	774840978	129140163				15083778125
12	262144	5898240	60604416	376012800	1568885760	4637606400	9958489920	15651921600	17870589360	14455195200	7863201036	2582803260	387420489			75418890625
13	1048576	25952256	295501824	2049490944	9659658240	32670397440	81572417280	152234095680	212349651840	218656085040	161549560944	81100022364	24794911296	3486784401		980445578125
14	2097152	55050240	668860416	4985487360	25467789312	94319769600	261156026880	549185443200	881401590720	1074361125600	979067377008	646848727560	292889889684	81358302690	10460353203	4902227890625

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Technical details

Node id: n-c7b786a59ccdf5e9 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-a14cd4083133b58e

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:     4,      9;
  2:     8,     30,      27;
  3:    32,    192,     378,     243;
  4:    64,    480,    1332,    1620,      729;
  5:   256,   2496,    9648,   18468,    17496,     6561;
  6:   512,   5760,   26784,   65880,    90396,    65610,    19683;
  7:  2048,  27648,  158976,  504576,   954504,  1076004,   669222,   177147;
  8:  4096,  61440,  400896, 1486080,  3422736,  5015520,  4566456,  2361960,   531441;
  9: 16384, 282624, 2156544, 9552384, 27065664, 50866704, 63405504, 50545944, 23383404, 4782969;

%F Formula

The row polynomials satisfy P(n,x) = (8+30*x+27*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[(8 + 30*x + 27*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] = 1; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 30*x + 27*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-0bd3c438ff845b55
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-0bd3c438ff845b55
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-0bd3c438ff845b55

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 A013610 e-0bd3c438ff845b55

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.