A038320 → 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

\(P_{n}(x) = \left(17 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 204\,x + 36\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	289	204	36								529
1	4913	5202	1836	216							12167
2	83521	117912	62424	14688	1296						279841
3	1419857	2505630	1768680	624240	110160	7776					6436343
4	24137569	51114852	45101340	21224160	5618160	793152	46656				148035889
5	410338673	1013777898	1073411892	631418760	222853680	47192544	5552064	279936			3404825447
6	6975757441	19696256304	24330669552	17174590272	7577025120	2139395328	377540352	38071296	1679616		78310985281
7	118587876497	376690901814	531798920208	437952051936	231856968672	81831871296	19254557952	2912454144	256981248	10077696	1801152661463

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6ffee7a877aa82b8 · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-dde9c6f2b382eac4

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 ?

A009967: 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:            289,             204,              36;
  1:           4913,            5202,            1836,             216;
  2:          83521,          117912,           62424,           14688,            1296;
  3:        1419857,         2505630,         1768680,          624240,          110160,           7776;
  4:       24137569,        51114852,        45101340,        21224160,         5618160,         793152,          46656;
  5:      410338673,      1013777898,      1073411892,       631418760,       222853680,       47192544,        5552064,        279936;
  6:     6975757441,     19696256304,     24330669552,     17174590272,      7577025120,     2139395328,      377540352,      38071296,       1679616;
  7:   118587876497,    376690901814,    531798920208,    437952051936,    231856968672,    81831871296,    19254557952,    2912454144,     256981248,     10077696;
  8:  2015993900449,   7115272589820,  11300727054420,  10635978404160,   6569280779040,  2782283624064,   818318712960,  165039068160,   21843406080,   1713208320,    60466176;
  9: 34271896307633, 133055597429634, 234803995464060, 248615995197240, 175493643668640, 86714506283328, 30605119864704, 7715576436480, 1361572312320, 160184977920, 11307174912, 362797056;

%F Formula

The row polynomials satisfy P(n,x) = (17+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 204*x + 36*x^2.

%t Mathematica

P[0, x_] := 289 + 204*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 6*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] = 289 + 204*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 6*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-a19af80736e38760
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-a19af80736e38760
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-a19af80736e38760

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-a19af80736e38760

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-940e6d9217c9d73c root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - basis_map A013609 e-2fddb004b6f97d1c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-da334b7c58401d78 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-c5656743efdffc45 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - parent_provers periodic_product e-de8595a50a8969d6 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003