A033282 → 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

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 22\,x^{2} + 16\,x^{3} + 4\,x^{4} + n + 3\,n x - n x^{2} - 7\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 18\,x^{3} + 14\,x^{4} + 4\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	3							4
2	1	7	11						19
3	1	12	42	45					100
4	1	18	104	242	197				562
5	1	25	210	790	1375	903			3304
6	1	33	375	2015	5595	7773	4279		20071
7	1	42	616	4410	17290	37968	43876	20793	124996

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-5acf616616571743 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cb8a225300b7095e

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 ?

A007564: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,  3;
  2: 1,  7,   11;
  3: 1, 12,   42,    45;
  4: 1, 18,  104,   242,    197;
  5: 1, 25,  210,   790,   1375,    903;
  6: 1, 33,  375,  2015,   5595,   7773,    4279;
  7: 1, 42,  616,  4410,  17290,  37968,   43876,   20793;
  8: 1, 52,  952,  8680,  44884, 138040,  250432,  247684,  103049;
  9: 1, 63, 1404, 15792, 103068, 415044, 1048152, 1618956, 1399293, 518859;

%F Formula

The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (n+2+(3*n+12)*x-(n-22)*x^2-(7*n-16)*x^3-(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (3*n + 12)*x - (n - 22)*x^2 - (7*n - 16)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 + (3*n + 12)*x - (n - 22)*x^2 - (7*n - 16)*x^3 - (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*x^2)); 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-67bfd12562549cc2
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-67bfd12562549cc2
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-67bfd12562549cc2

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-67bfd12562549cc2

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-1, 0)
  - theorem_check three_term_recurrence e-5bc6f2a826329589 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

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

A126216 → basis A007318

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