A038321 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 7\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 324 + 252\,x + 49\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	324	252	49								625
1	5832	6804	2646	343							15625
2	104976	163296	95256	24696	2401						390625
3	1889568	3674160	2857680	1111320	216090	16807					9765625
4	34012224	79361856	77157360	40007520	11668860	1815156	117649				244140625
5	612220032	1666598976	1944365472	1260236880	490092120	114354828	14823774	823543			6103515625
6	11019960576	34284321792	46664771328	36294822144	17643316320	5489031744	1067311728	118590192	5764801		152587890625
7	198359290368	694257516288	1079956136448	979960197888	571643448768	222305785632	57634833312	9605805552	933897762	40353607	3814697265625

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-32cf9641286d8e9f · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-c53984b4362c13dd

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 ?

A009969: 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,             252,              49;
  1:           5832,            6804,            2646,             343;
  2:         104976,          163296,           95256,           24696,            2401;
  3:        1889568,         3674160,         2857680,         1111320,          216090,           16807;
  4:       34012224,        79361856,        77157360,        40007520,        11668860,         1815156,          117649;
  5:      612220032,      1666598976,      1944365472,      1260236880,       490092120,       114354828,        14823774,         823543;
  6:    11019960576,     34284321792,     46664771328,     36294822144,     17643316320,      5489031744,      1067311728,      118590192,       5764801;
  7:   198359290368,    694257516288,   1079956136448,    979960197888,    571643448768,    222305785632,     57634833312,     9605805552,     933897762,     40353607;
  8:  3570467226624,  13885150325760,  24299013070080,  25198976517120,  17149303463040,   8003008282752,   2593567499040,   576348333120,   84050798580,   7263649260,   282475249;
  9: 64268410079232, 274925976450048, 534578287541760, 623674668798720, 485080297954560, 264099273330816, 102705272961984, 28529242489440, 5547352706280, 719101276740, 55930099302, 1977326743;

%F Formula

The row polynomials satisfy P(n,x) = (18+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 252*x + 49*x^2.

%t Mathematica

P[0, x_] := 324 + 252*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 7*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 + 252*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 7*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-5a47242ff4b92549
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-5a47242ff4b92549
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-5a47242ff4b92549

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 A007318 e-5a47242ff4b92549

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 ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A013613 → basis A013610

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