A046739 → γ

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 1 + 5\,x\)

\(P_{2}(x) = 1 + 18\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	1	5								6
2	1	18								19
3	1	47	61							109
4	1	108	479							588
5	1	233	2414	1385						4033
6	1	486	9970	19028						29485
7	1	995	36867	157658	50521					246042
8	1	2016	127401	1025268	1073517					2228203
9	1	4061	421372	5780619	13253431	2702765				22162249
10	1	8154	1353284	29707752	124815323	82112518				237997032
11	1	16343	4259417	143254820	996302182	1413861649	199360981			2757055393
12	1	32724	13219267	660115928	7116477278	18201001872	8200548715			34191395785
13	1	65489	40622874	2942301773	46972831682	195315316742	187817776972	19391512145		452480427678
14	1	131022	123963942	12793172844	292502841626	1849429517084	3168463002692	1037611984488		6360924613699
15	1	262091	376411711	54590486958	1742810449463	15990155023862	43993896574298	30504576100580	2404879675441	94691284984405

Computed in your browser ?

Technical details

Node id: n-c10753d48ae8e3dd · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-b7a7a452bc772ee8

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: 1,    5;
   2: 1,   18;
   3: 1,   47,      61;
   4: 1,  108,     479;
   5: 1,  233,    2414,     1385;
   6: 1,  486,    9970,    19028;
   7: 1,  995,   36867,   157658,     50521;
   8: 1, 2016,  127401,  1025268,   1073517;
   9: 1, 4061,  421372,  5780619,  13253431,  2702765;
  10: 1, 8154, 1353284, 29707752, 124815323, 82112518;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) + (n+2)*x*P(n-2,x) for n >= 3, with P(1,x) = 1 + 5*x, P(2,x) = 1 + 18*x.

%t Mathematica

P[1, x_] := 1 + 5*x;
P[2, x_] := 1 + 18*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x] + (n + 2)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 5*x; v[2] = 1 + 18*x; for(n = 3, N, my(i = n); v[i] = (1 + 2*n*x)*v[i - 1] + (x - 4*x^2)*deriv(v[i - 1], x) + (n + 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, 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-ecadc04944221db5
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-ecadc04944221db5
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-ecadc04944221db5

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-ecadc04944221db5

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-460746d5de499dd7 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