A110441 → 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

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(4 + 20\,x + 36\,x^{2} + 28\,x^{3} + 8\,x^{4} + 2\,n - 14\,n x^{2} - 20\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}(x) + \left(1 + 11\,x + 35\,x^{2} + 49\,x^{3} + 32\,x^{4} + 8\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	7	20	14						41
3	15	68	100	48					231
4	31	196	450	448	164				1289
5	63	516	1632	2512	1892	560			7175
6	127	1284	5214	10976	12704	7696	1912		39913
7	255	3076	15324	41200	64920	60192	30496	6528	221991

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-cf288f0202bf1453 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-d0d4468560fae926

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 ?

A152268: 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:    7,    20,     14;
  3:   15,    68,    100,     48;
  4:   31,   196,    450,    448,     164;
  5:   63,   516,   1632,   2512,    1892,     560;
  6:  127,  1284,   5214,  10976,   12704,    7696,    1912;
  7:  255,  3076,  15324,  41200,   64920,   60192,   30496,    6528;
  8:  511,  7172,  42458, 139520,  279820,  352096,  272240,  118528,  22288;
  9: 1023, 16388, 112600, 438544, 1072252, 1713104, 1793504, 1189120, 453872, 76096;

%F Formula

The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = (2*n+4+20*x-(14*n-36)*x^2-(20*n-28)*x^3-(8*n-8)*x^4)*P(n-1,x) + (1+11*x+35*x^2+49*x^3+32*x^4+8*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 + 4 + 20*x - (14*n - 36)*x^2 - (20*n - 28)*x^3 - (8*n - 8)*x^4)*P[n - 1, x] + (1 + 11*x + 35*x^2 + 49*x^3 + 32*x^4 + 8*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 + 4 + 20*x - (14*n - 36)*x^2 - (20*n - 28)*x^3 - (8*n - 8)*x^4)*v[i - 1] + (1 + 11*x + 35*x^2 + 49*x^3 + 32*x^4 + 8*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, 0)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-813d9e0ebd3ed0a7
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-813d9e0ebd3ed0a7
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-813d9e0ebd3ed0a7

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-813d9e0ebd3ed0a7

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array formed by the Mersenne numbers.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, 0)
  - theorem_check three_term_recurrence e-02d1e2cf79c791bd 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