A125662 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, −1/2) · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(8 + 36\,x + 60\,x^{2} + 44\,x^{3} + 12\,x^{4} - 2\,n - 16\,n x - 38\,n x^{2} - 36\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 31\,x + 75\,x^{2} + 89\,x^{3} + 52\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	8	22	15						45
3	21	88	122	56					287
4	55	310	651	604	209				1829
5	144	1020	2874	4028	2809	780			11655
6	377	3216	11376	21364	22471	12554	2911		74269
7	987	9848	41934	98800	139130	117120	54580	10864	473263

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-64bf2ebeb9f166a2 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-63c829d69f89a5ff

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 ?

A190973: 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:    3,     4;
  2:    8,    22,     15;
  3:   21,    88,    122,      56;
  4:   55,   310,    651,     604,     209;
  5:  144,  1020,   2874,    4028,    2809,     780;
  6:  377,  3216,  11376,   21364,   22471,   12554,    2911;
  7:  987,  9848,  41934,   98800,  139130,  117120,   54580,   10864;
  8: 2584, 29522, 147009,  416804,  736015,  829020,  581730,  232536,  40545;
  9: 6765, 87080, 496498, 1645932, 3496597, 4936940, 4633320, 2787384, 975471, 151316;

%F Formula

The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(2*n-8+(16*n-36)*x+(38*n-60)*x^2+(36*n-44)*x^3+(12*n-12)*x^4)*P(n-1,x) + (5+31*x+75*x^2+89*x^3+52*x^4+12*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[(-(2*n - 8 + (16*n - 36)*x + (38*n - 60)*x^2 + (36*n - 44)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*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] = (-(2*n - 8 + (16*n - 36)*x + (38*n - 60)*x^2 + (36*n - 44)*x^3 + (12*n - 12)*x^4)*v[i - 1] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*n*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, -1/2)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-1db48f91012bc01d
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-1db48f91012bc01d
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-1db48f91012bc01d

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-1db48f91012bc01d

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)
  - theorem_check three_term_recurrence e-c7df247a48c8552e root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

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