A207823 → D+1

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:      1;
  1:      5,       1;
  2:     23,      10,       1;
  3:    102,      70,      15,      1;
  4:    441,     418,     141,     20,      1;
  5:   1871,    2275,    1060,    236,     25,     1;
  6:   7823,   11644,    6966,   2140,    355,    30,    1;
  7:  32332,   57036,   41814,  16490,   3770,   498,   35,   1;
  8: 132369,  270300,  235110, 113644,  33335,  6062,  665,  40,  1;
  9: 537689, 1248357, 1258164, 721820, 259469, 60501, 9128, 856, 45, 1;

The row polynomials satisfy (2*n^3-2*n-10*n*x-2*n*x^2)*P(n,x) = (4*n^3+12*n^2-4*n-12+(n^3+3*n^2-26*n-28)*x-(10*n+10)*x^2-(n+1)*x^3)*P(n-1,x) + (12*n^2+12*n-96+(8*n^2+8*n-148)*x+(n^2+n-76)*x^2-15*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[((4*n^3 + 12*n^2 - 4*n - 12 + (n^3 + 3*n^2 - 26*n - 28)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (12*n^2 + 12*n - 96 + (8*n^2 + 8*n - 148)*x + (n^2 + n - 76)*x^2 - 15*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 2*n - 10*n*x - 2*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] = ((4*n^3 + 12*n^2 - 4*n - 12 + (n^3 + 3*n^2 - 26*n - 28)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*v[i - 1] + (12*n^2 + 12*n - 96 + (8*n^2 + 8*n - 148)*x + (n^2 + n - 76)*x^2 - 15*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 2*n - 10*n*x - 2*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-0c4a23d28a74f214
• 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-0c4a23d28a74f214
• 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-0c4a23d28a74f214
• 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-0c4a23d28a74f214

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-0c4a23d28a74f214

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

  • premise (sequence): Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-720cd5df5ac0e3b3 root: theorem check

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

    • preserver_image affine, shift 0 e-56c0c02612a8191c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)
        • theorem_check three_term_recurrence e-c7df247a48c8552e root: theorem check

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

    • theorem_check three_term_recurrence e-ec629e395a3def0c root: theorem check

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

    • basis_map A007318 e-9c2f7b8109ebb5e1

      Geometric basis map (elementary) polytool ?

      • premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-af8f298ca8f41adf root: theorem check

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

        • ↵ theorem_check three_term_recurrence e-c7df247a48c8552e (expanded above)

Related sequences ?

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. [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