A038314 → 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

\(P_{n}(x) = \left(22 + 12\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 528\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	528	144								1156
1	10648	17424	9504	1728							39304
2	234256	511104	418176	152064	20736						1336336
3	5153632	14055360	15333120	8363520	2280960	248832					45435424
4	113379904	371061504	505992960	367994880	150543360	32845824	2985984				1544804416
5	2494357888	9523911936	15584583168	14167802880	7727892480	2529128448	459841536	35831808			52523350144
6	54875873536	239458357248	457147772928	498706661376	340027269120	148375535616	40466055168	6306398208	429981696		1785793904896
7	1207269217792	5926594341888	12930751291392	16457319825408	13465079857152	7344589012992	2670759641088	624333422592	85136375808	5159780352	60716992766464

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b536b63a5727c887 · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-57ca34e86b523c55

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 ?

A009978: 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:             484,              528,              144;
  1:           10648,            17424,             9504,              1728;
  2:          234256,           511104,           418176,            152064,             20736;
  3:         5153632,         14055360,         15333120,           8363520,           2280960,            248832;
  4:       113379904,        371061504,        505992960,         367994880,         150543360,          32845824,          2985984;
  5:      2494357888,       9523911936,      15584583168,       14167802880,        7727892480,        2529128448,        459841536,         35831808;
  6:     54875873536,     239458357248,     457147772928,      498706661376,      340027269120,      148375535616,      40466055168,       6306398208,       429981696;
  7:   1207269217792,    5926594341888,   12930751291392,    16457319825408,    13465079857152,     7344589012992,    2670759641088,     624333422592,     85136375808,      5159780352;
  8:  26559922791424,  144872306135040,  355595660513280,   517230051655680,   493719594762240,   323161916571648,  146891780259840,   45784450990080,   9365001338880,   1135151677440,    61917364224;
  9: 584318301411328, 3505909808467968, 9561572204912640, 15646209062584320, 17068591704637440, 13034197301723136, 7109562164576256, 2769959284899840, 755443441336320, 137353352970240, 14984002142208, 743008370688;

%F Formula

The row polynomials satisfy P(n,x) = (22+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 528*x + 144*x^2.

%t Mathematica

P[0, x_] := 484 + 528*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 12*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 + 528*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 12*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-f9a8e29d679c9a8f
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-f9a8e29d679c9a8f
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-f9a8e29d679c9a8f

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-f9a8e29d679c9a8f

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 (3)

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

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