Real-rooted atlas

A060920 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence
\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-11 - 52\,x - 89\,x^{2} - 66\,x^{3} - 18\,x^{4} + 13\,n + 59\,n x + 97\,n x^{2} + 69\,n x^{3} + 18\,n x^{4}\right)\,P_{n-1}(x) + \left(-5 - 34\,x - 90\,x^{2} - 116\,x^{3} - 73\,x^{4} - 18\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
25151131
313598843203
4342074684651711345
5896802063310423146838933
623321428157164621855911075273159359
7610655530042760951150051036655158010923394475

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-af29b8b59eec534e · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-bfdae63ac656708a

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:    1;
  1:    2,     3;
  2:    5,    15,     11;
  3:   13,    59,     88,      43;
  4:   34,   207,    468,     465,     171;
  5:   89,   680,   2063,    3104,    2314,     683;
  6:  233,  2142,   8157,   16462,   18559,   11075,    2731;
  7:  610,  6555,  30042,   76095,  115005,  103665,   51580,   10923;
  8: 1597, 19641, 105244,  320837,  608455,  734861,  551798,  235445,   43691;
  9: 4181, 57913, 355235, 1266248, 2889997, 4378793, 4403439, 2833422, 1058286, 174763;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (13*n-11+(59*n-52)*x+(97*n-89)*x^2+(69*n-66)*x^3+(18*n-18)*x^4)*P(n-1,x) - (5+34*x+90*x^2+116*x^3+73*x^4+18*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[((13*n - 11 + (59*n - 52)*x + (97*n - 89)*x^2 + (69*n - 66)*x^3 + (18*n - 18)*x^4)*P[n - 1, x] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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] = ((13*n - 11 + (59*n - 52)*x + (97*n - 89)*x^2 + (69*n - 66)*x^3 + (18*n - 18)*x^4)*v[i - 1] - (5 + 34*x + 90*x^2 + 116*x^3 + 73*x^4 + 18*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*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 (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-2d8eb3c5691a59ac

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