A038314 → basis A013611

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(22 + 48\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 2112\,x + 2304\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	2112	2304								4900
1	10648	69696	152064	110592							343000
2	234256	2044416	6690816	9732096	5308416						24010000
3	5153632	56221440	245329920	535265280	583925760	254803968					1680700000
4	113379904	1484246016	8095887360	23551672320	38539100160	33634123776	12230590464				117649000000
5	2494357888	38095647744	249353330688	906739384320	1978340474880	2589827530752	1883510931456	587068342272			8235430000000
6	54875873536	957833428992	7314364366848	31917226328064	87046980894720	151936548470784	165748961968128	103324028239872	28179280429056		576480100000000
7	1207269217792	23706377367552	206892020662272	1053268468826112	3447060443430912	7520859149303808	10939431489896448	10229078795747328	5579497524953088	1352605460594688	40353607000000000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-9b691fa7ebf67931 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-83fc6524b6311133

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,              2112,               2304;
  1:           10648,             69696,             152064,              110592;
  2:          234256,           2044416,            6690816,             9732096,             5308416;
  3:         5153632,          56221440,          245329920,           535265280,           583925760,            254803968;
  4:       113379904,        1484246016,         8095887360,         23551672320,         38539100160,          33634123776,          12230590464;
  5:      2494357888,       38095647744,       249353330688,        906739384320,       1978340474880,        2589827530752,        1883510931456,         587068342272;
  6:     54875873536,      957833428992,      7314364366848,      31917226328064,      87046980894720,      151936548470784,      165748961968128,      103324028239872,       28179280429056;
  7:   1207269217792,    23706377367552,    206892020662272,    1053268468826112,    3447060443430912,     7520859149303808,    10939431489896448,    10229078795747328,     5579497524953088,     1352605460594688;
  8:  26559922791424,   579489224540160,   5689530568212480,   33102723305963520,  126392216259133440,   330917802569367552,   601668731944304640,   750132445021470720,   613744727744839680,   297573201330831360,    64925062108545024;
  9: 584318301411328, 14023639233871872, 152985155278602240, 1001357380005396480, 4369559476387184640, 13347018036964491264, 29120766626104344576, 45383012923798978560, 49508741371417067520, 36006357361030594560, 15711865030267895808, 3116402981210161152;

%F Formula

The row polynomials satisfy P(n,x) = (22+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 2112*x + 2304*x^2.

%t Mathematica

P[0, x_] := 484 + 2112*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 48*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 + 2112*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 48*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-6524eda715a9c2e7
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-6524eda715a9c2e7
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-6524eda715a9c2e7

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 A013611 e-6524eda715a9c2e7

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.

A038226 → basis A013610

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