A257610 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 - 6\,x + 6\,n x\right)\,P_{n-1}(x) + \left(3\,x - 12\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	12							16
3	8	108							116
4	16	684	648						1348
5	32	3804	13392						17228
6	64	19980	175608	80352					276004
7	128	102204	1884384	2991168					4977884
8	256	516396	18141192	66821760	17947008				103426612
9	512	2594268	163719792	1170427968	1053119232				2389861772
10	1024	12998988	1418717592	17786301408	35811372672	6318715392			61348107076
11	2048	65056380	11973692160	246723148800	928230451200	537154633728			1724146984316
12	4096	325417068	99302581800	3216849707520	20396920780800	25839776894976	3222927802368		52776103188628
13	8192	1627380252	813945678480	40151870709120	401363480401920	928802305953792	374535878787072		1745669108918828
14	16384	8137540236	6618972524472	485623644438240	7305467295409920	27830543613272064	24209159082909696	2247215272722432	62084636018833444

Computed in your browser ?

Technical details

Node id: n-26452d815810c4a7 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-03e551e467821f7e

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:   2;
  2:   4,      12;
  3:   8,     108;
  4:  16,     684,       648;
  5:  32,    3804,     13392;
  6:  64,   19980,    175608,      80352;
  7: 128,  102204,   1884384,    2991168;
  8: 256,  516396,  18141192,   66821760,   17947008;
  9: 512, 2594268, 163719792, 1170427968, 1053119232;

%F Formula

The row polynomials satisfy P(n,x) = (2+(6*n-6)*x)*P(n-1,x) + (3*x-12*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + (6*n - 6)*x)*P[n - 1, x] + (3*x - 12*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (6*n - 6)*x)*v[i - 1] + (3*x - 12*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 - 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-a7cdd8e861c7d98f
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-a7cdd8e861c7d98f
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-a7cdd8e861c7d98f

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-a7cdd8e861c7d98f

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-be1c7e65de7af5de 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