A257180 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,n x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 9 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
1	9	6				15
2	27	60				87
3	81	402	120			603
4	243	2256	2208			4707
5	729	11454	24576	4416		41175
6	2187	54564	214512	124800		396063
7	6561	248874	1618200	2035872	249600	4159107
8	19683	1100472	11077488	25160832	9890688	47249163

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-9fdebfaf3bddb35b · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-6ebcca3d5bcd5855

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:      9,        6;
   2:     27,       60;
   3:     81,      402,       120;
   4:    243,     2256,      2208;
   5:    729,    11454,     24576,       4416;
   6:   2187,    54564,    214512,     124800;
   7:   6561,   248874,   1618200,    2035872,     249600;
   8:  19683,  1100472,  11077488,   25160832,    9890688;
   9:  59049,  4756182,  70794048,  261739872,  220199808,   19781376;
  10: 177147, 20205708, 430069152, 2419967808, 3635317632, 1039050240;

%F Formula

The row polynomials satisfy P(n,x) = (3+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 6*x.

%t Mathematica

P[1, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 9 + 6*x; for(n = 2, N, my(i = n); v[i] = (3 + 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-418ae03de1c699ba
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-418ae03de1c699ba
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-418ae03de1c699ba

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-418ae03de1c699ba

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-27a325e06cca4d67 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

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