A038331 → 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

\(P_{n}(x) = \left(17 + 15\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 510\,x + 225\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	289	510	225								1024
1	4913	13005	11475	3375							32768
2	83521	294780	390150	229500	50625						1048576
3	1419857	6264075	11054250	9753750	4303125	759375					33554432
4	24137569	127787130	281883375	331627500	219459375	77456250	11390625				1073741824
5	410338673	2534444745	6708824325	9865918125	8705221875	4608646875	1355484375	170859375			34359738368
6	6975757441	49240640760	152066684700	268352973000	295977543750	208925325000	92172937500	23236875000	2562890625		1099511627776
7	118587876497	941727254535	3323743251300	6843000811500	9056912838750	7991393681250	4700819812500	1777620937500	392122265625	38443359375	35184372088832

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8a857e84342335b8 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-08a7d3764c1b9874

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 ?

A009976: 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:            289,             510,              225;
  1:           4913,           13005,            11475,             3375;
  2:          83521,          294780,           390150,           229500,            50625;
  3:        1419857,         6264075,         11054250,          9753750,          4303125,           759375;
  4:       24137569,       127787130,        281883375,        331627500,        219459375,         77456250,         11390625;
  5:      410338673,      2534444745,       6708824325,       9865918125,       8705221875,       4608646875,       1355484375,        170859375;
  6:     6975757441,     49240640760,     152066684700,     268352973000,     295977543750,     208925325000,      92172937500,      23236875000,       2562890625;
  7:   118587876497,    941727254535,    3323743251300,    6843000811500,    9056912838750,    7991393681250,    4700819812500,    1777620937500,     392122265625,     38443359375;
  8:  2015993900449,  17788181474550,   70629544090125,  166187162565000,  256612530431250,  271707385162500,  199784842031250,  100731853125000,   33330392578125,   6535371093750,    576650390625;
  9: 34271896307633, 332638993574085, 1467524971650375, 3884624924956875, 6855220455806250, 8468213504231250, 7471953091968750, 4709214133593750, 2077594470703125, 611057197265625, 107833623046875, 8649755859375;

%F Formula

The row polynomials satisfy P(n,x) = (17+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 510*x + 225*x^2.

%t Mathematica

P[0, x_] := 289 + 510*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 15*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] = 289 + 510*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 15*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-633275fbd6b7b077
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-633275fbd6b7b077
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-633275fbd6b7b077

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-633275fbd6b7b077

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-49423b91d6bf9513 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b8d4501ecbbc70e3 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