A391533 → x ↦ x/(x+1)

nonnegative coefficients only rational roots

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

Recurrence

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

\(P_{0}(x) = 4 + 12\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	12	9															25
1	4	20	33	18														75
2	4	28	73	84	36													225
3	4	36	129	230	204	72												675
4	4	44	201	488	664	480	144											2025
5	4	52	289	890	1640	1808	1104	288										6075
6	4	60	393	1468	3420	5088	4720	2496	576									18225
7	4	68	513	2254	6356	11928	14896	11936	5568	1152								54675
8	4	76	649	3280	10864	24640	38752	41728	29440	12288	2304							164025
9	4	84	801	4578	17424	46368	88032	119232	112896	71168	26880	4608						492075
10	4	92	969	6180	26580	81216	180768	295296	351360	296960	169216	58368	9216					1476225
11	4	100	1153	8118	38940	134376	343200	656832	941952	999680	763136	396800	125952	18432				4428675
12	4	108	1353	10424	55176	212256	611952	1343232	2255616	2883584	2762496	1923072	919552	270336	36864			13286025
13	4	116	1569	13130	76024	322608	1036464	2567136	4942080	7394816	8529664	7448064	4765696	2109440	577536	73728		39858075
14	4	124	1801	16268	102284	474656	1681680	4640064	10076352	17278976	23319296	24507392	19661824	11640832	4796416	1228800	147456	119574225

Computed in your browser ?

Technical details

Node id: n-c84985983df5f062 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-efbd87188225e7d4

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

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: 4, 12,   9;
  1: 4, 20,  33,   18;
  2: 4, 28,  73,   84,    36;
  3: 4, 36, 129,  230,   204,    72;
  4: 4, 44, 201,  488,   664,   480,   144;
  5: 4, 52, 289,  890,  1640,  1808,  1104,    288;
  6: 4, 60, 393, 1468,  3420,  5088,  4720,   2496,    576;
  7: 4, 68, 513, 2254,  6356, 11928, 14896,  11936,   5568,  1152;
  8: 4, 76, 649, 3280, 10864, 24640, 38752,  41728,  29440, 12288,  2304;
  9: 4, 84, 801, 4578, 17424, 46368, 88032, 119232, 112896, 71168, 26880, 4608;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+3*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 4 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((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] = 4 + 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((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-360765530d5f86fe
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-360765530d5f86fe
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-360765530d5f86fe

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-360765530d5f86fe

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check
    
    Parent-row provers of polytool polytool ?

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