Real-rooted atlas

A207606 → 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
\(\left(18 + 36\,x + 18\,x^{2} + 24\,n + 48\,n x + 24\,n x^{2} + 8\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-6 - 57\,x - 153\,x^{2} - 159\,x^{3} - 57\,x^{4} - 4\,n - 53\,n x - 144\,n x^{2} - 130\,n x^{3} - 20\,n x^{4} + 15\,n x^{5} + 5\,n^{2} + 15\,n^{2} x + 22\,n^{2} x^{2} + 34\,n^{2} x^{3} + 37\,n^{2} x^{4} + 15\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(18 + 123\,x + 318\,x^{2} + 396\,x^{3} + 240\,x^{4} + 57\,x^{5} - 8\,n x - 58\,n x^{2} - 156\,n x^{3} - 200\,n x^{4} - 124\,n x^{5} - 30\,n x^{6}\right)\,P_{n-1}'(x) + \left(12\,x + 75\,x^{2} + 195\,x^{3} + 270\,x^{4} + 210\,x^{5} + 87\,x^{6} + 15\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 4 + 15\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04151332
15315834128
265416220189512
37853617076552332048
481257011918277820526108192
59175123944258880101886255159732768
61023620449122237203749035582186874181131072
711309319817298558371137671481231198255497410946524288

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-a9766ae9536fc7a1 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d049d83a04ac9840

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:  4,  15,   13;
  1:  5,  31,   58,    34;
  2:  6,  54,  162,   201,     89;
  3:  7,  85,  361,   707,    655,    233;
  4:  8, 125,  701,  1918,   2778,   2052,     610;
  5:  9, 175, 1239,  4425,   8880,  10188,    6255,    1597;
  6: 10, 236, 2044,  9122,  23720,  37490,   35582,   18687,    4181;
  7: 11, 309, 3198, 17298,  55837, 113767,  148123,  119825,   54974,   10946;
  8: 12, 395, 4797, 30744, 119560, 300993,  503265,  556678,  392286,  159765,  28657;
  9: 13, 495, 6952, 51876, 237721, 717927, 1478015, 2093313, 2011859, 1255669, 459743, 75025;
%F Formula
The row polynomials satisfy (8*n^2+24*n+18+(16*n^2+48*n+36)*x+(8*n^2+24*n+18)*x^2)*P(n,x) = (5*n^2-4*n-6+(15*n^2-53*n-57)*x+(22*n^2-144*n-153)*x^2+(34*n^2-130*n-159)*x^3+(37*n^2-20*n-57)*x^4+(15*n^2+15*n)*x^5)*P(n-1,x) + (18-(8*n-123)*x-(58*n-318)*x^2-(156*n-396)*x^3-(200*n-240)*x^4-(124*n-57)*x^5-30*n*x^6)*d/dx P(n-1,x) + (12*x+75*x^2+195*x^3+270*x^4+210*x^5+87*x^6+15*x^7)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4 + 15*x + 13*x^2.
%t Mathematica
P[0, x_] := 4 + 15*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 4*n - 6 + (15*n^2 - 53*n - 57)*x + (22*n^2 - 144*n - 153)*x^2 + (34*n^2 - 130*n - 159)*x^3 + (37*n^2 - 20*n - 57)*x^4 + (15*n^2 + 15*n)*x^5)*P[n - 1, x] + (18 - (8*n - 123)*x - (58*n - 318)*x^2 - (156*n - 396)*x^3 - (200*n - 240)*x^4 - (124*n - 57)*x^5 - 30*n*x^6)*D[P[n - 1, x], x] + (12*x + 75*x^2 + 195*x^3 + 270*x^4 + 210*x^5 + 87*x^6 + 15*x^7)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 15*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^2 - 4*n - 6 + (15*n^2 - 53*n - 57)*x + (22*n^2 - 144*n - 153)*x^2 + (34*n^2 - 130*n - 159)*x^3 + (37*n^2 - 20*n - 57)*x^4 + (15*n^2 + 15*n)*x^5)*v[i - 1] + (18 - (8*n - 123)*x - (58*n - 318)*x^2 - (156*n - 396)*x^3 - (200*n - 240)*x^4 - (124*n - 57)*x^5 - 30*n*x^6)*deriv(v[i - 1], x) + (12*x + 75*x^2 + 195*x^3 + 270*x^4 + 210*x^5 + 87*x^6 + 15*x^7)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*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-f3e51c43b80b3409
  • 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-f3e51c43b80b3409
  • 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-f3e51c43b80b3409

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