A038331 → 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

\(P_{n}(x) = \left(17 + 10\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 340\,x + 100\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	289	340	100								729
1	4913	8670	5100	1000							19683
2	83521	196520	173400	68000	10000						531441
3	1419857	4176050	4913000	2890000	850000	100000					14348907
4	24137569	85191420	125281500	98260000	43350000	10200000	1000000				387420489
5	410338673	1689629830	2981699700	2923235000	1719550000	606900000	119000000	10000000			10460353203
6	6975757441	32827093840	67585193200	79511992000	58464700000	27512800000	8092000000	1360000000	100000000		282429536481
7	118587876497	627818169690	1477219222800	2027555796000	1789019820000	1052364600000	412692000000	104040000000	15300000000	1000000000	7625597484987

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-688fdf685007c036 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-700950e0f523054a

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 ?

A009971: 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,             340,             100;
  1:           4913,            8670,            5100,             1000;
  2:          83521,          196520,          173400,            68000,            10000;
  3:        1419857,         4176050,         4913000,          2890000,           850000,           100000;
  4:       24137569,        85191420,       125281500,         98260000,         43350000,         10200000,         1000000;
  5:      410338673,      1689629830,      2981699700,       2923235000,       1719550000,        606900000,       119000000,        10000000;
  6:     6975757441,     32827093840,     67585193200,      79511992000,      58464700000,      27512800000,      8092000000,      1360000000,      100000000;
  7:   118587876497,    627818169690,   1477219222800,    2027555796000,    1789019820000,    1052364600000,    412692000000,    104040000000,    15300000000,     1000000000;
  8:  2015993900449,  11858787649700,  31390908484500,   49240640760000,   50688894900000,   35780396400000,  17539410000000,   5895600000000,  1300500000000,   170000000000,   10000000000;
  9: 34271896307633, 221759329049390, 652233320733500, 1150999977765000, 1354117620900000, 1115155687800000, 655973934000000, 275619300000000, 81064500000000, 15895000000000, 1870000000000, 100000000000;

%F Formula

The row polynomials satisfy P(n,x) = (17+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 340*x + 100*x^2.

%t Mathematica

P[0, x_] := 289 + 340*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 10*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 + 340*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 10*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-2f341f062a4ee4be
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-2f341f062a4ee4be
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-2f341f062a4ee4be

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-2f341f062a4ee4be

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