A166340 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 2\,n x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	1	6								7
2	1	16								17
3	1	38	32							71
4	1	84	248							333
5	1	178	1248	496						1923
6	1	368	5168	6976						12513
7	1	750	19184	58912	13952					92799
8	1	1516	66552	389120	305408					762597
9	1	3050	220880	2222000	3861760	610816				6918507
10	1	6120	711440	11538560	37084800	19111936				68452857
11	1	12262	2244480	56114400	300809600	337180416	38223872			734585031
12	1	24548	6978680	260369280	2177420800	4429559296	1616288768			8490641373
13	1	49122	21476096	1167093360	14532273920	48351563776	37891377152	3232577536		105196410963
14	1	98272	65607216	5097895360	91334863360	464496669696	652052150272	177426128896		1390473413073
15	1	196574	199376720	21834940192	548436433280	4065668105216	9209331748864	5331721932800	354852257792	19532044991439

Computed in your browser ?

Technical details

Node id: n-7da7145fe72dc08d · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-eae0d43706dd3f1f

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 1)
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): no

OEIS matches ?

No OEIS matches.

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,    6;
   2: 1,   16;
   3: 1,   38,     32;
   4: 1,   84,    248;
   5: 1,  178,   1248,      496;
   6: 1,  368,   5168,     6976;
   7: 1,  750,  19184,    58912,    13952;
   8: 1, 1516,  66552,   389120,   305408;
   9: 1, 3050, 220880,  2222000,  3861760,   610816;
  10: 1, 6120, 711440, 11538560, 37084800, 19111936;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 6*x.

%t Mathematica

P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 6*x; for(n = 2, N, my(i = n); v[i] = (1 + 2*n*x)*v[i - 1] + (x - 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-b61cfbcaf1cd183b
Coefficients of one sign. Every nonzero row, n >= 1: 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-b61cfbcaf1cd183b
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-b61cfbcaf1cd183b

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

gamma_polynomial scale 1, shift 0 e-b61cfbcaf1cd183b

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-7ffc7a4c532b7fe3 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-9eddb58049332ffc 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