Real-rooted atlas

A207607 → 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
\(\left(8 + 16\,x + 8\,x^{2} - 16\,n - 32\,n x - 16\,n x^{2} + 8\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(20 + 84\,x + 154\,x^{2} + 166\,x^{3} + 106\,x^{4} + 30\,x^{5} - 25\,n - 99\,n x - 176\,n x^{2} - 200\,n x^{3} - 143\,n x^{4} - 45\,n x^{5} + 5\,n^{2} + 15\,n^{2} x + 22\,n^{2} x^{2} + 34\,n^{2} x^{3} + 37\,n^{2} x^{4} + 15\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(6 + 68\,x + 268\,x^{2} + 516\,x^{3} + 530\,x^{4} + 280\,x^{5} + 60\,x^{6} - 8\,n x - 58\,n x^{2} - 156\,n x^{3} - 200\,n x^{4} - 124\,n x^{5} - 30\,n x^{6}\right)\,P_{n-1}'(x) + \left(12\,x + 75\,x^{2} + 195\,x^{3} + 270\,x^{4} + 210\,x^{5} + 87\,x^{6} + 15\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
217816
3112302164
41187310955256
51251453443651441024
6133255850141611643774096
7142413180641845358359398716384

Show all 15 stored rows

Generated b-files: A000302 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-106df0aaa5357879 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-429c048c3a049f47

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,  3;
  2: 1,  7,   8;
  3: 1, 12,  30,   21;
  4: 1, 18,  73,  109,    55;
  5: 1, 25, 145,  344,   365,   144;
  6: 1, 33, 255,  850,  1416,  1164,   377;
  7: 1, 42, 413, 1806,  4184,  5358,  3593,   987;
  8: 1, 52, 630, 3458, 10415, 18422, 19139, 10835,  2584;
  9: 1, 63, 918, 6132, 22995, 52557, 75051, 65556, 32106, 6765;
%F Formula
The row polynomials satisfy (8*n^2-16*n+8+(16*n^2-32*n+16)*x+(8*n^2-16*n+8)*x^2)*P(n,x) = (5*n^2-25*n+20+(15*n^2-99*n+84)*x+(22*n^2-176*n+154)*x^2+(34*n^2-200*n+166)*x^3+(37*n^2-143*n+106)*x^4+(15*n^2-45*n+30)*x^5)*P(n-1,x) + (6-(8*n-68)*x-(58*n-268)*x^2-(156*n-516)*x^3-(200*n-530)*x^4-(124*n-280)*x^5-(30*n-60)*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 >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 25*n + 20 + (15*n^2 - 99*n + 84)*x + (22*n^2 - 176*n + 154)*x^2 + (34*n^2 - 200*n + 166)*x^3 + (37*n^2 - 143*n + 106)*x^4 + (15*n^2 - 45*n + 30)*x^5)*P[n - 1, x] + (6 - (8*n - 68)*x - (58*n - 268)*x^2 - (156*n - 516)*x^3 - (200*n - 530)*x^4 - (124*n - 280)*x^5 - (30*n - 60)*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 - 16*n + 8 + (16*n^2 - 32*n + 16)*x + (8*n^2 - 16*n + 8)*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; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n^2 - 25*n + 20 + (15*n^2 - 99*n + 84)*x + (22*n^2 - 176*n + 154)*x^2 + (34*n^2 - 200*n + 166)*x^3 + (37*n^2 - 143*n + 106)*x^4 + (15*n^2 - 45*n + 30)*x^5)*v[i - 1] + (6 - (8*n - 68)*x - (58*n - 268)*x^2 - (156*n - 516)*x^3 - (200*n - 530)*x^4 - (124*n - 280)*x^5 - (30*n - 60)*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 - 16*n + 8 + (16*n^2 - 32*n + 16)*x + (8*n^2 - 16*n + 8)*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-ba8938d63a27da5b
  • 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-ba8938d63a27da5b
  • 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-ba8938d63a27da5b

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