A053117 → D+1

OEIS matches ?

• 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 ?
• A006527: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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

The row polynomials satisfy (n^3-3*n^2+3*n-1+(3*n-3)*x-(n-1)*x^2)*P(n,x) = (n+n^3*x+2*n*x^2-n*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 >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + n^3*x + 2*n*x^2 - n*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^2 + 3*n - 1 + (3*n - 3)*x - (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n + n^3*x + 2*n*x^2 - n*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^2 + 3*n - 1 + (3*n - 3)*x - (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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-df5dd69f4658c66c
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-df5dd69f4658c66c

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-df5dd69f4658c66c

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

  • premise (sequence): Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check three_term_recurrence e-b99a39d00579baec root: theorem check

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

    • theorem_check three_term_recurrence e-dc563681f97c05c6 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.

• chebyshev_u → 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