A038330 → 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

\(P_{n}(x) = \left(16 + 12\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 256 + 384\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	256	384	144								784
1	4096	9216	6912	1728							21952
2	65536	196608	221184	110592	20736						614656
3	1048576	3932160	5898240	4423680	1658880	248832					17210368
4	16777216	75497472	141557760	141557760	79626240	23887872	2985984				481890304
5	268435456	1409286144	3170893824	3963617280	2972712960	1337720832	334430208	35831808			13492928512
6	4294967296	25769803776	67645734912	101468602368	95126814720	57076088832	21403533312	4586471424	429981696		377801998336
7	68719476736	463856467968	1391569403904	2435246456832	2739652263936	2054739197952	1027369598976	330225942528	61917364224	5159780352	10578455953408

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-fcf922cdd7b5dfe1 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-c91d5d506c107735

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 ?

A009972: 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:            256,             384,             144;
  1:           4096,            9216,            6912,             1728;
  2:          65536,          196608,          221184,           110592,            20736;
  3:        1048576,         3932160,         5898240,          4423680,          1658880,           248832;
  4:       16777216,        75497472,       141557760,        141557760,         79626240,         23887872,          2985984;
  5:      268435456,      1409286144,      3170893824,       3963617280,       2972712960,       1337720832,        334430208,        35831808;
  6:     4294967296,     25769803776,     67645734912,     101468602368,      95126814720,      57076088832,      21403533312,      4586471424,       429981696;
  7:    68719476736,    463856467968,   1391569403904,    2435246456832,    2739652263936,    2054739197952,    1027369598976,    330225942528,     61917364224,     5159780352;
  8:  1099511627776,   8246337208320,  27831388078080,   55662776156160,   73057393704960,   65751654334464,   41094783959040,  17612050268160,   4953389137920,   825564856320,    61917364224;
  9: 17592186044416, 145135534866432, 544258255749120, 1224581075435520, 1836871613153280, 1928715193810944, 1446536395358208, 774930211799040, 290598829424640, 72649707356160, 10897456103424, 743008370688;

%F Formula

The row polynomials satisfy P(n,x) = (16+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 384*x + 144*x^2.

%t Mathematica

P[0, x_] := 256 + 384*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 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] = 256 + 384*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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-77aaa200c8a95097
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-77aaa200c8a95097
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-77aaa200c8a95097

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-77aaa200c8a95097

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-016885d52a801213 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-0a35d6a32dc0856a root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (6)

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