A038329 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 225 + 180\,x + 36\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	225	180	36								441
1	3375	4050	1620	216							9261
2	50625	81000	48600	12960	1296						194481
3	759375	1518750	1215000	486000	97200	7776					4084101
4	11390625	27337500	27337500	14580000	4374000	699840	46656				85766121
5	170859375	478406250	574087500	382725000	153090000	36741600	4898880	279936			1801088541
6	2562890625	8201250000	11481750000	9185400000	4592700000	1469664000	293932800	33592320	1679616		37822859361
7	38443359375	138396093750	221433750000	206671500000	124002900000	49601160000	13226976000	2267481600	226748160	10077696	794280046581

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4ef858fd88b37fab · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9629d1e9c04dbfbe

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 ?

A009965: 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:           225,            180,             36;
  1:          3375,           4050,           1620,            216;
  2:         50625,          81000,          48600,          12960,           1296;
  3:        759375,        1518750,        1215000,         486000,          97200,           7776;
  4:      11390625,       27337500,       27337500,       14580000,        4374000,         699840,          46656;
  5:     170859375,      478406250,      574087500,      382725000,      153090000,       36741600,        4898880,        279936;
  6:    2562890625,     8201250000,    11481750000,     9185400000,     4592700000,     1469664000,      293932800,      33592320,      1679616;
  7:   38443359375,   138396093750,   221433750000,   206671500000,   124002900000,    49601160000,    13226976000,    2267481600,    226748160,     10077696;
  8:  576650390625,  2306601562500,  4151882812500,  4428675000000,  3100072500000,  1488034800000,   496011600000,  113374080000,  17006112000,   1511654400,   60466176;
  9: 8649755859375, 38058925781250, 76117851562500, 91341421875000, 73073137500000, 40920957000000, 16368382800000, 4676680800000, 935336160000, 124711488000, 9976919040, 362797056;

%F Formula

The row polynomials satisfy P(n,x) = (15+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 180*x + 36*x^2.

%t Mathematica

P[0, x_] := 225 + 180*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 6*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] = 225 + 180*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 6*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-f3e6c91e6b1bdc2d
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-f3e6c91e6b1bdc2d
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-f3e6c91e6b1bdc2d

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 A013609 e-f3e6c91e6b1bdc2d

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-d1ea7de7e1f61f8b root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-a6c3b06cf2d0ed0d 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