Real-rooted atlas

A123519 → 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
\(\left(3 + 6\,x + 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(3 + 9\,x + 3\,x^{2} - 9\,x^{3} - 6\,x^{4} + 2\,n + 4\,n x - 4\,n x^{2} - 12\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 18\,x^{2} + 30\,x^{3} + 22\,x^{4} + 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 8\,x + 11\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0181120
11154741104
2124126240153544
313527083411375712848
4148505223248515144213114912
51638615075155552601322559795378080
618013721030441590971681319009673629681408832
7199207619236978302984985623326419884078411107712140672

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f356d4c20400a3cf · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-935c57c7e92571a4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 ?

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,   8,   11;
  1: 1,  15,   47,    41;
  2: 1,  24,  126,   240,    153;
  3: 1,  35,  270,   834,   1137,     571;
  4: 1,  48,  505,  2232,   4851,    5144,    2131;
  5: 1,  63,  861,  5075,  15555,   26013,   22559,     7953;
  6: 1,  80, 1372, 10304,  41590,   97168,  131900,    96736,    29681;
  7: 1,  99, 2076, 19236,  97830,  298498,  562332,   641988,   407841,   110771;
  8: 1, 120, 3015, 33648, 209034,  797664, 1950830,  3075600,  3028581,  1696808,  413403;
  9: 1, 143, 4235, 55869, 414282, 1918070, 5835158, 11889370, 16105413, 13939499, 6984655, 1542841;
%F Formula
The row polynomials satisfy (2*n+3+(4*n+6)*x+(2*n+3)*x^2)*P(n,x) = (2*n+3+(4*n+9)*x-(4*n-3)*x^2-(12*n+9)*x^3-(6*n+6)*x^4)*P(n-1,x) + (4*x+18*x^2+30*x^3+22*x^4+6*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 11*x^2.
%t Mathematica
P[0, x_] := 1 + 8*x + 11*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 3 + (4*n + 9)*x - (4*n - 3)*x^2 - (12*n + 9)*x^3 - (6*n + 6)*x^4)*P[n - 1, x] + (4*x + 18*x^2 + 30*x^3 + 22*x^4 + 6*x^5)*D[P[n - 1, x], x])/(2*n + 3 + (4*n + 6)*x + (2*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] = 1 + 8*x + 11*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 3 + (4*n + 9)*x - (4*n - 3)*x^2 - (12*n + 9)*x^3 - (6*n + 6)*x^4)*v[i - 1] + (4*x + 18*x^2 + 30*x^3 + 22*x^4 + 6*x^5)*deriv(v[i - 1], x))/(2*n + 3 + (4*n + 6)*x + (2*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-10dce9fcf56455e3
  • 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-10dce9fcf56455e3
  • 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-10dce9fcf56455e3

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

No candidate equalities or related sequences.

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