A079510 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

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

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	2														2
2	0	3	9													12
3	0	0	20	44												64
4	0	0	15	160	265											440
5	0	0	0	210	1344	1854										3408
6	0	0	0	105	2695	12383	14833									30016
7	0	0	0	0	2520	33992	124648	133496								294656
8	0	0	0	0	945	47880	441630	1366776	1334961							3192192
9	0	0	0	0	0	34650	843920	6002700	16249200	14684570						37815040
10	0	0	0	0	0	10395	918225	14667730	85922166	208460307	176214841					486193664
11	0	0	0	0	0	0	540540	21561540	258172200	1298088792	2872728924	2290792932				6741884928
12	0	0	0	0	0	0	135135	19099080	482917435	4663907248	20702126445	42346252936	32071101049			100285539328
13	0	0	0	0	0	0	0	9459450	576816240	10677116450	87082355360	348221218254	665136847088	481066515734		1592770328576
14	0	0	0	0	0	0	0	2027025	430404975	16182491325	237359116995	1686725835795	6169043964525	11093313016095	7697064251745	26900121108480

Technical details

Node id: n-064d20684eb59b67 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-3f24d40b8bba79ee

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, 2;
  2: 0, 3,  9;
  3: 0, 0, 20,  44;
  4: 0, 0, 15, 160,  265;
  5: 0, 0,  0, 210, 1344,  1854;
  6: 0, 0,  0, 105, 2695, 12383,  14833;
  7: 0, 0,  0,   0, 2520, 33992, 124648,  133496;
  8: 0, 0,  0,   0,  945, 47880, 441630, 1366776,  1334961;
  9: 0, 0,  0,   0,    0, 34650, 843920, 6002700, 16249200, 14684570;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*P[n - 2, x] + ((n + 1)*x + (n + 1)*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*v[i - 2] + ((n + 1)*x + (n + 1)*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 Triangle T(n,k) read by rows; related to number of preorders.?
  - 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 ?

A008306 → basis A007318

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