A144439 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 4\,x + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 8\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	6							7
3	1	28							29
4	1	98	124						223
5	1	312	1460						1773
6	1	958	11240	6088					18287
7	1	2900	72152	135456					210509
8	1	8730	420676	1836496	554000				2819903
9	1	26224	2318700	19730592	19948880				42024397
10	1	78710	12345232	185329496	419980016	80903520			698636975
11	1	236172	64297328	1598229440	6784553264	4289676608			12736992813
12	1	708562	330146252	13012621728	93396227168	129441041888	17320513472		253501259071
13	1	2125736	1679546084	101781626816	1156065424864	2931760201984	1269274363456		5460563288941
14	1	6377262	8492679928	773595339240	13280499618080	55557463173440	51940571232512	5111738480768	126672366901231

Computed in your browser ?

Technical details

Node id: n-4aa60654988fe331 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-ade87970e32786bf

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,     6;
  3: 1,    28;
  4: 1,    98,     124;
  5: 1,   312,    1460;
  6: 1,   958,   11240,     6088;
  7: 1,  2900,   72152,   135456;
  8: 1,  8730,  420676,  1836496,   554000;
  9: 1, 26224, 2318700, 19730592, 19948880;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-4)*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 4)*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x] + 2*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + (4*n - 4)*x)*v[i - 1] + (2*x - 8*x^2)*deriv(v[i - 1], x) + 2*x*v[i - 2]); 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-9b31724594a81849
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-9b31724594a81849
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-9b31724594a81849

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-9b31724594a81849

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-b81aae17045623cf root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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