A159854 → 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

\(P_{n}(x) = \left(2 + 3\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 6\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	6	9															16
1	2	15	36	27														80
2	4	36	117	162	81													400
3	8	84	342	675	648	243												2000
4	16	192	936	2376	3321	2430	729											10000
5	32	432	2448	7560	13770	14823	8748	2187										50000
6	64	960	6192	22464	50220	70956	61965	30618	6561									250000
7	128	2112	15264	63504	167832	292572	336798	247131	104976	19683								1250000
8	256	4608	36864	172800	526176	1088640	1551312	1504656	951345	354294	59049							6250000
9	512	9984	87552	456192	1570752	3755808	6368544	7663248	6416658	3562623	1180980	177147						31250000
10	1024	21504	205056	1175040	4510080	12223872	24004512	34432128	35823060	26375220	13049829	3897234	531441					156250000
11	2048	46080	474624	2965248	12545280	37977984	84680640	140877792	174942504	160219620	105225318	46943955	12754584	1594323				781250000
12	4096	98304	1087488	7354368	33986304	113591808	283295232	535797504	772518384	845266752	691109496	409563864	166341033	41452398	4782969			3906250000
13	8192	208896	2469888	17971200	90035712	329142528	907365888	1921480704	3152429280	4008088656	3918019248	2892456216	1561373658	581927895	133923132	14348907		19531250000
14	16384	442368	5566464	43352064	233985024	928392192	2802159360	6565059072	12069300672	17473465152	19860304464	17538970176	11800115964	5847976764	2013629949	430467210	43046721	97656250000

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Technical details

Node id: n-ca8f77e380f6014f · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-60d64fb2105f3f8c

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 ?

A055842: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    6,     9;
  1:   2,   15,    36,     27;
  2:   4,   36,   117,    162,      81;
  3:   8,   84,   342,    675,     648,     243;
  4:  16,  192,   936,   2376,    3321,    2430,     729;
  5:  32,  432,  2448,   7560,   13770,   14823,    8748,    2187;
  6:  64,  960,  6192,  22464,   50220,   70956,   61965,   30618,    6561;
  7: 128, 2112, 15264,  63504,  167832,  292572,  336798,  247131,  104976,   19683;
  8: 256, 4608, 36864, 172800,  526176, 1088640, 1551312, 1504656,  951345,  354294,   59049;
  9: 512, 9984, 87552, 456192, 1570752, 3755808, 6368544, 7663248, 6416658, 3562623, 1180980, 177147;

%F Formula

The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2.

%t Mathematica

P[0, x_] := 1 + 6*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 3*x)*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 + 6*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*x)*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-02e979b2d9fb24b7
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-02e979b2d9fb24b7
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-02e979b2d9fb24b7

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-02e979b2d9fb24b7

Geometric basis map (elementary) polytool ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.