A227450 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 5\,x + 5\,x^{2} + x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	2	6														8
2	2	16	22													40
3	2	34	110	94												240
4	2	68	392	748	454											1664
5	2	134	1236	3964	5226	2430										12992
6	2	264	3698	17792	38278	38008	14214									112256
7	2	522	10818	73546	229974	366062	288998	89918								1059840
8	2	1036	31396	291052	1242872	2822084	3529436	2299204	610182							10827264
9	2	2062	91064	1125560	6317036	19205860	33844984	34633432	19125602	4412798						118758400
10	2	4112	264926	4303616	30947732	121115872	282153148	403313920	347662922	166117136	33827974					1389711360
11	2	8210	774198	16382678	148315636	727517652	2152733260	4031697740	4823551146	3580195898	1504070366	273646526				17258893312
12	2	16404	2273312	62351820	701845042	4234044488	15485958672	36499409048	56838480454	58263577188	37876079568	14172210908	2326980998			226463227904
13	2	32790	6705084	237860276	3299436630	24141255506	107020994856	308814836984	601127617294	798008089818	713655148124	411912674580	138737335642	20732504062		3127694491648
14	2	65560	19852906	910817120	15471986122	135870057016	719452034322	2491430641536	5896260939590	9733957580040	11228633261438	8887471216928	4605572013934	1408752406696	192982729350	45316785602560

Computed in your browser ?

Technical details

Node id: n-2781c74a5cedd697 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-5344ee972d4f418a

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 ?

A055882: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 2;
  1: 2,    6;
  2: 2,   16,    22;
  3: 2,   34,   110,      94;
  4: 2,   68,   392,     748,     454;
  5: 2,  134,  1236,    3964,    5226,     2430;
  6: 2,  264,  3698,   17792,   38278,    38008,    14214;
  7: 2,  522, 10818,   73546,  229974,   366062,   288998,    89918;
  8: 2, 1036, 31396,  291052, 1242872,  2822084,  3529436,  2299204,   610182;
  9: 2, 2062, 91064, 1125560, 6317036, 19205860, 33844984, 34633432, 19125602, 4412798;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1-(n-5)*x-(2*n-5)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (n - 5)*x - (2*n - 5)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((1 - (n - 5)*x - (2*n - 5)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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 [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-12a33f06274c1738
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-12a33f06274c1738
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-12a33f06274c1738

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

preserver_image mobius, shift 0 e-12a33f06274c1738

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-74361f30d2bc482d 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