chebyshev_u → D+1

OEIS matches ?

• A006527: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167875: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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.

Triangle begins:
  0:  1;
  1:  2,   2;
  2: -1,   8,    4;
  3: -4,  -4,   24,    8;
  4:  1, -24,  -12,   64,   16;
  5:  6,   6,  -96,  -32,  160,    32;
  6: -1,  48,   24, -320,  -80,   384,    64;
  7: -8,  -8,  240,   80, -960,  -192,   896,   128;
  8:  1, -80,  -40,  960,  240, -2688,  -448,  2048,  256;
  9: 10,  10, -480, -160, 3360,   672, -7168, -1024, 4608, 512;

The row polynomials satisfy (n^3+3*n*x-n*x^2)*P(n,x) = (n+1+(n^3+3*n^2+3*n+1)*x+(2*n+2)*x^2-(n+1)*x^3)*P(n-1,x) - (n^2+n+1+x-(n^2+n+2)*x^2-x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (n^3 + 3*n^2 + 3*n + 1)*x + (2*n + 2)*x^2 - (n + 1)*x^3)*P[n - 1, x] - (n^2 + n + 1 + x - (n^2 + n + 2)*x^2 - x^3 + x^4)*D[P[n - 1, x], x])/(n^3 + 3*n*x - n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (n^3 + 3*n^2 + 3*n + 1)*x + (2*n + 2)*x^2 - (n + 1)*x^3)*v[i - 1] - (n^2 + n + 1 + x - (n^2 + n + 2)*x^2 - x^3 + x^4)*deriv(v[i - 1], x))/(n^3 + 3*n*x - n*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)

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-4219cb192b8f36cb
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-4219cb192b8f36cb

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

• preserver_image derivative_polynomial, shift 0 e-4219cb192b8f36cb

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): chebyshev_u
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check three_term_recurrence e-a7b3a0c81265a049 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-8ab839cedecfc233 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

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

• A053117 → D+1

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023