A038333 → 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

\(P_{n}(x) = \left(19 + 14\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 532\,x + 196\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	361	532	196								1089
1	6859	15162	11172	2744							35937
2	130321	384104	424536	208544	38416						1185921
3	2476099	9122470	13443640	9905840	3649520	537824					39135393
4	47045881	207992316	383143740	376421920	208022640	61311936	7529536				1291467969
5	893871739	4610496338	10191623484	12516028840	9222337040	4077243744	1001428288	105413504			42618442977
6	16983563041	100113634768	258187794928	380487276736	350448807520	206580349696	76108549888	16022852608	1475789056		1406408618241
7	322687697779	2139928943166	6307158990384	10843887386976	11985349217184	8831309949504	4338187343616	1369953897984	252359928576	20661046784	46411484401953

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-649aa8398fb69113 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b35a687a1de5c177

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 ?

A009977: 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:             361,             532,              196;
  1:            6859,           15162,            11172,             2744;
  2:          130321,          384104,           424536,           208544,             38416;
  3:         2476099,         9122470,         13443640,          9905840,           3649520,            537824;
  4:        47045881,       207992316,        383143740,        376421920,         208022640,          61311936,          7529536;
  5:       893871739,      4610496338,      10191623484,      12516028840,        9222337040,        4077243744,       1001428288,        105413504;
  6:     16983563041,    100113634768,     258187794928,     380487276736,      350448807520,      206580349696,      76108549888,      16022852608,       1475789056;
  7:    322687697779,   2139928943166,    6307158990384,   10843887386976,    11985349217184,     8831309949504,    4338187343616,    1369953897984,     252359928576,     20661046784;
  8:   6131066257801,  45176277689060,  149795026021620,  294334086217920,   379536058544160,   335589778081152,  206063898821760,   86763746872320,   23974193214720,   3925598888960,   289254654976;
  9: 116490258898219, 944184203701354, 3478573382057620, 7689478002443160, 11331862319389920, 11689710603160128, 8613470970749568, 4533405774078720, 1670202127292160, 410225083896320, 60454222889984, 4049565169664;

%F Formula

The row polynomials satisfy P(n,x) = (19+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 532*x + 196*x^2.

%t Mathematica

P[0, x_] := 361 + 532*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 14*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] = 361 + 532*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 14*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-20f4338b8f3f0a8d
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-20f4338b8f3f0a8d
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-20f4338b8f3f0a8d

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-20f4338b8f3f0a8d

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-de9af033a7745ebd root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-400776ae4466efdf 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