A131222 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-4 - 12\,x - 12\,x^{2} - 4\,x^{3} + 6\,n + 18\,n x + 18\,n x^{2} + 6\,n x^{3} - 2\,n^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-2}(x) + \left(-3 - 5\,x - 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	3	4													7
3	0	14	37	24												75
4	0	90	353	454	192											1089
5	0	744	3846	7349	6166	1920										20025
6	0	7560	48274	121771	151954	93936	23040									446535
7	0	91440	692532	2161984	3566221	3281914	1599144	322560								11715795
8	0	1285200	11229756	41663532	85177609	103729396	75303388	30193584	5160960							353743425
9	0	20603520	203598000	873158012	2124586164	3210395649	3086846596	1845321372	627345072	92897280						12084751665
10	0	371226240	4086822096	19857223188	55930492988	100701362091	120252606648	95283096868	48324697056	14239656000	1857945600					460905128775
11	0	7428153600	90047769120	488235824856	1560078674628	3254995467338	4635520093821	4565029873788	3070646720428	1350531705456	350802570240	40874803200				19414191656475
12	0	163459296000	2161693196640	12924945773928	46141626786628	109325219683406	180573127745625	212225941128818	177530788046116	103611662439000	40188932779104	9325935509760	980995276800			895154327661825
13	0	3923502105600	56174272882560	366880183729344	1445899661280504	3831013498365052	7191390045503522	9809298307630821	9798704379104642	7115649280203892	3664089737073624	1270169115806304	266181669192960	25505877196800		44844879530075625
14	0	102017281766400	1571289848588160	11122819144225920	47929492141402728	140311086149913108	294750811212731158	457216641931059011	530363994098908254	460137953038201868	294943240694622376	135785760385216800	42523448419245312	8120163046625280	714164561510400	2425592881954016775

Technical details

Node id: n-7a42f3124cafc007 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cbe3741e35d7b661

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0,        1;
  2: 0,        3,         4;
  3: 0,       14,        37,        24;
  4: 0,       90,       353,       454,        192;
  5: 0,      744,      3846,      7349,       6166,       1920;
  6: 0,     7560,     48274,    121771,     151954,      93936,      23040;
  7: 0,    91440,    692532,   2161984,    3566221,    3281914,    1599144,     322560;
  8: 0,  1285200,  11229756,  41663532,   85177609,  103729396,   75303388,   30193584,   5160960;
  9: 0, 20603520, 203598000, 873158012, 2124586164, 3210395649, 3086846596, 1845321372, 627345072, 92897280;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = -(2*n^2-6*n+4+(6*n^2-18*n+12)*x+(6*n^2-18*n+12)*x^2+(2*n^2-6*n+4)*x^3)*P(n-2,x) + (3*n-3+(6*n-5)*x+(3*n-2)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 6*n + 4 + (6*n^2 - 18*n + 12)*x + (6*n^2 - 18*n + 12)*x^2 + (2*n^2 - 6*n + 4)*x^3)*P[n - 2, x] + (3*n - 3 + (6*n - 5)*x + (3*n - 2)*x^2)*P[n - 1, 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] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 6*n + 4 + (6*n^2 - 18*n + 12)*x + (6*n^2 - 18*n + 12)*x^2 + (2*n^2 - 6*n + 4)*x^3)*v[i - 2] + (3*n - 3 + (6*n - 5)*x + (3*n - 2)*x^2)*v[i - 1])/(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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A079638 → x ↦ x/(x+1)

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