A249248 → 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

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

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	9	11													21
3	1	15	49	38												103
4	1	22	129	236	131											519
5	1	30	269	878	1125	490										2793
6	1	39	490	2473	5436	5309	1897									15645
7	1	49	816	5845	19558	32205	25435	7714								91623
8	1	60	1274	12238	57915	142502	186550	123438	32501							556479
9	1	72	1894	23436	149001	512362	989184	1072938	610867	142142						3501897
10	1	85	2709	41898	344505	1582047	4218495	6670878	6166611	3079205	641795					22748229
11	1	99	3755	70908	732261	4340301	15318945	33126420	44253831	35618087	15836313	2989294				152290215
12	1	114	5071	114740	1453968	10828062	49044723	139171356	252068520	291013106	207268095	83034920	14321923			1048324599
13	1	130	6699	178838	2728730	24980364	141716715	512490246	1208646342	1878473660	1907501159	1217932710	443950261	70493962		7409099817
14	1	147	8684	270011	4883571	53968444	375907893	1694015451	5056868019	10164364484	13810672869	12507153043	7234360906	2418747129	355816769	53677037421

Computed in your browser ?

Technical details

Node id: n-3659b3de32876d6c · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-783f05b943b24c5b

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 ?

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,  4;
  2: 1,  9,   11;
  3: 1, 15,   49,    38;
  4: 1, 22,  129,   236,    131;
  5: 1, 30,  269,   878,   1125,    490;
  6: 1, 39,  490,  2473,   5436,   5309,   1897;
  7: 1, 49,  816,  5845,  19558,  32205,  25435,    7714;
  8: 1, 60, 1274, 12238,  57915, 142502, 186550,  123438,  32501;
  9: 1, 72, 1894, 23436, 149001, 512362, 989184, 1072938, 610867, 142142;

%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) + (1+3*x+2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*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] + (1 + 3*x + 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] = 1 + 4*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] + (1 + 3*x + 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)

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-8d9c6b636eca438b
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-8d9c6b636eca438b
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-8d9c6b636eca438b

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-8d9c6b636eca438b

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-2dcae0cd92b4b84a 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