Real-rooted atlas

A108838 → 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
\(\left(3 + 6\,x + 3\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 21\,x + 24\,x^{2} + 9\,x^{3} + 5\,n + 19\,n x + 23\,n x^{2} + 9\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1358
24161434
35357042152
4664216288132706
57105525115511554293376
681601100352057204576143016514
7923120799009210212702718018486282256

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f7cfaa6a3770b14b · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5cd009bfc7bc0bb1

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 ?

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:  2;
  1:  3,   5;
  2:  4,  16,   14;
  3:  5,  35,   70,    42;
  4:  6,  64,  216,   288,    132;
  5:  7, 105,  525,  1155,   1155,    429;
  6:  8, 160, 1100,  3520,   5720,   4576,   1430;
  7:  9, 231, 2079,  9009,  21021,  27027,  18018,   4862;
  8: 10, 320, 3640, 20384,  63700, 116480, 123760,  70720,  16796;
  9: 11, 429, 6006, 42042, 168168, 408408, 612612, 554268, 277134, 58786;
%F Formula
The row polynomials satisfy (n^2+4*n+3+(2*n^2+8*n+6)*x+(n^2+4*n+3)*x^2)*P(n,x) = (n^2+5*n+6+(4*n^2+19*n+21)*x+(5*n^2+23*n+24)*x^2+(2*n^2+9*n+9)*x^3)*P(n-1,x) + ((2*n+3)*x+(6*n+9)*x^2+(6*n+9)*x^3+(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (4*n^2 + 19*n + 21)*x + (5*n^2 + 23*n + 24)*x^2 + (2*n^2 + 9*n + 9)*x^3)*P[n - 1, x] + ((2*n + 3)*x + (6*n + 9)*x^2 + (6*n + 9)*x^3 + (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + 3 + (2*n^2 + 8*n + 6)*x + (n^2 + 4*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (4*n^2 + 19*n + 21)*x + (5*n^2 + 23*n + 24)*x^2 + (2*n^2 + 9*n + 9)*x^3)*v[i - 1] + ((2*n + 3)*x + (6*n + 9)*x^2 + (6*n + 9)*x^3 + (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + 3 + (2*n^2 + 8*n + 6)*x + (n^2 + 4*n + 3)*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-cbb4d6a46b756fe5
  • 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-cbb4d6a46b756fe5
  • 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-cbb4d6a46b756fe5

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

Same fingerprint (1)

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

References

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