A038314 → 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

\(P_{n}(x) = \left(22 + 36\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 1584\,x + 1296\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	1584	1296								3364
1	10648	52272	85536	46656							195112
2	234256	1533312	3763584	4105728	1679616						11316496
3	5153632	42166080	137998080	225815040	184757760	60466176					656356768
4	113379904	1113184512	4553936640	9935861760	12194012160	7981535232	2176782336				38068692544
5	2494357888	28571735808	140261248512	382530677760	625959290880	614578212864	335224479744	78364164096			2207984167552
6	54875873536	718375071744	4114329956352	13465079857152	27542208798720	36055255154688	29499754217472	13792092880896	2821109907456		128063081718016
7	1207269217792	17779783025664	116376761622528	444347635286016	1090671468429312	1784735130157056	1946983778353152	1365417195208704	558579761676288	101559956668416	7427658739644928

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d2377120f9386969 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-5e3baf07c55bade9

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 ?

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:             484,              1584,              1296;
  1:           10648,             52272,             85536,              46656;
  2:          234256,           1533312,           3763584,            4105728,             1679616;
  3:         5153632,          42166080,         137998080,          225815040,           184757760,            60466176;
  4:       113379904,        1113184512,        4553936640,         9935861760,         12194012160,          7981535232,          2176782336;
  5:      2494357888,       28571735808,      140261248512,       382530677760,        625959290880,        614578212864,        335224479744,         78364164096;
  6:     54875873536,      718375071744,     4114329956352,     13465079857152,      27542208798720,      36055255154688,      29499754217472,      13792092880896,       2821109907456;
  7:   1207269217792,    17779783025664,   116376761622528,    444347635286016,    1090671468429312,    1784735130157056,    1946983778353152,    1365417195208704,     558579761676288,     101559956668416;
  8:  26559922791424,   434616918405120,  3200360944619520,  13965211394703360,   39991287175741440,   78528345726910464,  107084107809423360,  100130594315304960,   61443773784391680,   22343190467051520,   3656158440062976;
  9: 584318301411328, 10517729425403904, 86054149844213760, 422447644689776640, 1382555928075632640, 3167309944318722048, 5182870817976090624, 6057900956075950080, 4956464418607595520, 2703526046513233920, 884790342495240192, 131621703842267136;

%F Formula

The row polynomials satisfy P(n,x) = (22+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1584*x + 1296*x^2.

%t Mathematica

P[0, x_] := 484 + 1584*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 36*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] = 484 + 1584*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 36*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-0be1d4b95988d4e0
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-0be1d4b95988d4e0
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-0be1d4b95988d4e0

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-0be1d4b95988d4e0

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-50ef4b08e4862a18 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-08ecf9f1e7a97c44 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A038277 → basis A007318

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