A082985 → D

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.

Triangle begins:
  0:  0;
  1:  3;
  2:  5,  10;
  3:  7,  28,   21;
  4:  9,  54,   90,   36;
  5: 11,  88,  231,  220,    55;
  6: 13, 130,  468,  728,   455,    78;
  7: 15, 180,  825, 1800,  1890,   840,  105;
  8: 17, 238, 1326, 3740,  5610,  4284, 1428,  136;
  9: 19, 304, 1995, 6916, 13585, 15048, 8778, 2280, 171;

The row polynomials satisfy (4*n^2-6*n+2)*P(n,x) = (4*n^2-2*n-2+(12*n^2-10*n-8)*x)*P(n-1,x) - ((2*n+1)*x+(8*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n - 2 + (12*n^2 - 10*n - 8)*x)*P[n - 1, x] - ((2*n + 1)*x + (8*n + 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n - 2 + (12*n^2 - 10*n - 8)*x)*v[i - 1] - ((2*n + 1)*x + (8*n + 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-7551b5ec3efde91d
• 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-7551b5ec3efde91d
• 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-7551b5ec3efde91d
• 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-7551b5ec3efde91d

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-7551b5ec3efde91d

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

  • premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4fa62233b528a616 root: theorem check

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

Related sequences ?

Same fingerprint (5)

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

• A078812 → reverse
• A172431 → reverse
• A091044 → basis A013611
• A172431: Even row Pascal-square read by antidiagonals.
• A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-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