A038321 → 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

\(P_{n}(x) = \left(18 + 28\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 324 + 1008\,x + 784\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	324	1008	784								2116
1	5832	27216	42336	21952							97336
2	104976	653184	1524096	1580544	614656						4477456
3	1889568	14696640	45722880	71124480	55319040	17210368					205962976
4	34012224	317447424	1234517760	2560481280	2987228160	1858719744	481890304				9474296896
5	612220032	6666395904	31109847552	80655160320	125463582720	117099343872	60718178304	13492928512			435817657216
6	11019960576	137137287168	746636341248	2322868617216	4516688977920	5620768505856	4371708837888	1942981705728	377801998336		20047612231936
7	198359290368	2777030065152	17279298183168	62717452664832	146340722884608	227641124487168	236072277245952	157381518163968	61203923730432	10578455953408	922190162669056

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4b2df01c2184142e · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-9111d60e4a92b32c

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 ?

A009990: 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:            324,             1008,              784;
  1:           5832,            27216,            42336,             21952;
  2:         104976,           653184,          1524096,           1580544,             614656;
  3:        1889568,         14696640,         45722880,          71124480,           55319040,           17210368;
  4:       34012224,        317447424,       1234517760,        2560481280,         2987228160,         1858719744,          481890304;
  5:      612220032,       6666395904,      31109847552,       80655160320,       125463582720,       117099343872,        60718178304,        13492928512;
  6:    11019960576,     137137287168,     746636341248,     2322868617216,      4516688977920,      5620768505856,      4371708837888,      1942981705728,       377801998336;
  7:   198359290368,    2777030065152,   17279298183168,    62717452664832,    146340722884608,    227641124487168,    236072277245952,    157381518163968,     61203923730432,     10578455953408;
  8:  3570467226624,   55540601303040,  388784209121280,  1612734497095680,   4390221686538240,   8195080481538048,  10623252476067840,   9442891089838080,   5508353135738880,   1904122071613440,   296196766695424;
  9: 64268410079232, 1099703905800192, 8553252600668160, 39915178803118080, 124180556276367360, 270437655890755584, 420680798052286464, 467423108946984960, 363551306958766080, 188508085089730560, 58646959805693952, 8293509467471872;

%F Formula

The row polynomials satisfy P(n,x) = (18+28*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 784*x^2.

%t Mathematica

P[0, x_] := 324 + 1008*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 28*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] = 324 + 1008*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 28*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-e460f89b751823ed
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-e460f89b751823ed
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-e460f89b751823ed

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-e460f89b751823ed

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-f6f8234ea600f6f8 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-16634379058ff4dd 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