A257622 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 - 6\,x + 6\,n x\right)\,P_{n-1}(x) + \left(3\,x - 12\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4								4
2	16	24							40
3	64	360							424
4	256	3672	2160						6088
5	1024	31848	65664						98536
6	4096	253656	1229904	393984					1881640
7	16384	1923048	18386784	19880640					40206856
8	65536	14149464	241559280	589410432	119283840				964468552
9	262144	102191976	2924973504	13459758336	8981466624				25468652584
10	1048576	729499608	33541798032	262726063488	385979116032	53888799744			736866325480
11	4194304	5169411816	370433961504	4622943554496	12481091380224	5655636587520			23135279089864
12	16777216	36462706776	3983487853104	75656492591616	338385768718464	332116740006912	33933819525120		784112788179208
13	67108864	256446906984	42022640937600	1174741820640000	8137806032793600	14431476509374464	4731944909635584		28518248427397096
14	268435456	1800362840280	437151905236944	17540866278950400	179544052991577600	518332234661922816	363869365180723200	28391669457813504	1108117141107500200

Computed in your browser ?

Technical details

Node id: n-962c510e0ddbead5 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-e71b5f5493602054

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:      4;
  2:     16,        24;
  3:     64,       360;
  4:    256,      3672,       2160;
  5:   1024,     31848,      65664;
  6:   4096,    253656,    1229904,      393984;
  7:  16384,   1923048,   18386784,    19880640;
  8:  65536,  14149464,  241559280,   589410432,  119283840;
  9: 262144, 102191976, 2924973504, 13459758336, 8981466624;

%F Formula

The row polynomials satisfy P(n,x) = (4+(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[(4 + (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] = (4 + (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-b55284588c13a182
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-b55284588c13a182
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-b55284588c13a182

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-b55284588c13a182

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 + 4.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-36659981c068b032 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