A082985 → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

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

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	11	15	5						31
3	29	56	35	7					127
4	76	189	171	66	9				511
5	199	605	715	407	110	11			2047
6	521	1872	2730	2054	832	169	13		8191
7	1364	5655	9810	9180	4965	1533	245	15	32767

Show all 15 stored rows

Generated b-files: A083420 (row sums), A200073 ?

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-4 - 3\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Technical details

Node id: n-0135f5d413e6cc58 · revision 2 · JSON
Definition source: derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-e3101e1faeaf859f)
Fingerprint: f6-347434f4a19cf0c5

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 ?

A267938: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A233868: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A083420: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A267941: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A267888: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A200073: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
A267886: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A267890: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A267935: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:    4,     3;
  2:   11,    15,      5;
  3:   29,    56,     35,      7;
  4:   76,   189,    171,     66,      9;
  5:  199,   605,    715,    407,    110,    11;
  6:  521,  1872,   2730,   2054,    832,   169,    13;
  7: 1364,  5655,   9810,   9180,   4965,  1533,   245,   15;
  8: 3571, 16779,  33745,  37774,  25585, 10642,  2618,  340,  17;
  9: 9349, 49096, 112309, 146357, 119168, 62453, 20862, 4218, 456, 19;

%F Formula

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

%t Mathematica

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

A200073: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:    1;
  1:    4,     3;
  2:   11,    15,      5;
  3:   29,    56,     35,      7;
  4:   76,   189,    171,     66,      9;
  5:  199,   605,    715,    407,    110,    11;
  6:  521,  1872,   2730,   2054,    832,   169,    13;
  7: 1364,  5655,   9810,   9180,   4965,  1533,   245,   15;
  8: 3571, 16779,  33745,  37774,  25585, 10642,  2618,  340,  17;
  9: 9349, 49096, 112309, 146357, 119168, 62453, 20862, 4218, 456, 19;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] + (3 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-d52b551add083e42
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-557cf165357d4e1f, e-d52b551add083e42
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-557cf165357d4e1f, e-d52b551add083e42
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-557cf165357d4e1f, e-d52b551add083e42
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-557cf165357d4e1f

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A007318 e-d52b551add083e42

Geometric basis map (elementary) polytool ?
- premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4fa62233b528a616 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
definition_equality e-e3101e1faeaf859f root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image affine, shift 0 e-557cf165357d4e1f

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ theorem_check three_term_recurrence e-4fa62233b528a616 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A111125 → x ↦ x/(x+1)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023