A257608 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 18\,x + 18\,n x\right)\,P_{n-1}(x) + \left(9\,x - 36\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	18							19
3	1	216							217
4	1	2214	3888						6103
5	1	22212	153576						175789
6	1	222210	4117392	2764368					7103971
7	1	2222208	94229568	225628416					322080193
8	1	22222206	1990360512	11405992320	4061311488				17479886527
9	1	222222204	40216847976	462673741824	560884248576				1063997060581
10	1	2222222202	792120109248	16574381088912	45737099255808	10095916474368			73201739150539
11	1	22222222200	15370282072800	549631642288320	2885628110866560	2110947731030016			5561599988479897
12	1	222222222198	295635359379600	17326341525245760	156235087908011520	252927513614174976	37997059158540288		464781859787574343
13	1	2222222222196	5657071828208040	527709054457543680	7651943137322968320	22883591955628878336	11195228743830014976		42264132185289835549
14	1	22222222222194	107924364735947568	15692299160980925520	349613236942600331520	1741320112317995552256	1851451546514610253824	201514117388940269568	4159699258912085502451

Computed in your browser ?

Technical details

Node id: n-9b7738e1007099e2 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-e9aca16ff8c15554

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,        18;
  3: 1,       216;
  4: 1,      2214,        3888;
  5: 1,     22212,      153576;
  6: 1,    222210,     4117392,      2764368;
  7: 1,   2222208,    94229568,    225628416;
  8: 1,  22222206,  1990360512,  11405992320,   4061311488;
  9: 1, 222222204, 40216847976, 462673741824, 560884248576;

%F Formula

The row polynomials satisfy P(n,x) = (1+(18*n-18)*x)*P(n-1,x) + (9*x-36*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[(1 + (18*n - 18)*x)*P[n - 1, x] + (9*x - 36*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] = (1 + (18*n - 18)*x)*v[i - 1] + (9*x - 36*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-96a51d1c5f8c0af4
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-96a51d1c5f8c0af4
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-96a51d1c5f8c0af4

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-96a51d1c5f8c0af4

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) = 9*x + 1.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-e4e9153fc21b93ec 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