A122431 → 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

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

\(P_{0}(x) = 8 + 36\,x + 54\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	P_n(1)
0	8	36	54	27															125
1	8	60	162	189	81														500
2	8	84	342	675	648	243													2000
3	8	108	594	1701	2673	2187	729												8000
4	8	132	918	3483	7776	10206	7290	2187											32000
5	8	156	1314	6237	18225	33534	37908	24057	6561										128000
6	8	180	1782	10179	36936	88209	138510	137781	78732	19683									512000
7	8	204	2322	15525	67473	199017	403137	553311	492075	255879	59049								2048000
8	8	228	2934	22491	114048	401436	1000188	1762722	2152008	1732104	826686	177147							8192000
9	8	252	3618	31293	181521	743580	2204496	4763286	7440174	8188128	6022998	2657205	531441						32768000
10	8	276	4374	42147	275400	1288143	4435236	11376774	21730032	30508650	30587382	20726199	8503056	1594323					131072000
11	8	300	5202	55269	401841	2114343	8299665	24682482	55860354	95698746	122113332	112488345	70681653	27103491	4782969				524288000
12	8	324	6102	70875	567648	3319866	14642694	49581477	129907800	263279808	409209570	478828341	408146688	239148450	86093442	14348907			2097152000
13	8	348	7074	89181	780273	5022810	24602292	93509559	278652231	653003208	1199048994	1706457051	1844631711	1463588514	803538792	272629233	43046721		8388608000
14	8	372	8118	110403	1047816	7363629	39670722	167316435	559180908	1488959901	3158058618	5303604033	6964002864	6997483647	5194304334	2683245609	860934420	129140163	33554432000

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Technical details

Node id: n-0482a458bbc39c21 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-ba483ebdff40bb6b

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
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: 8,  36,   54,    27;
  1: 8,  60,  162,   189,     81;
  2: 8,  84,  342,   675,    648,    243;
  3: 8, 108,  594,  1701,   2673,   2187,     729;
  4: 8, 132,  918,  3483,   7776,  10206,    7290,    2187;
  5: 8, 156, 1314,  6237,  18225,  33534,   37908,   24057,    6561;
  6: 8, 180, 1782, 10179,  36936,  88209,  138510,  137781,   78732,   19683;
  7: 8, 204, 2322, 15525,  67473, 199017,  403137,  553311,  492075,  255879,   59049;
  8: 8, 228, 2934, 22491, 114048, 401436, 1000188, 1762722, 2152008, 1732104,  826686,  177147;
  9: 8, 252, 3618, 31293, 181521, 743580, 2204496, 4763286, 7440174, 8188128, 6022998, 2657205, 531441;

%F Formula

The row polynomials satisfy P(n,x) = (1+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 36*x + 54*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 8 + 36*x + 54*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 8 + 36*x + 54*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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-02645aa9ee0a8704
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-02645aa9ee0a8704
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-02645aa9ee0a8704

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-02645aa9ee0a8704

Geometric basis map (elementary) polytool ?
- premise (sequence): Riordan array ((1+x)^3,x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-73f8c6c9e8d957af root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.