A207824 → basis A008277

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:       6,        1;
  2:      35,       13,        1;
  3:     204,      125,       21,        1;
  4:    1189,     1066,      292,       30,       1;
  5:    6930,     8526,     3381,      561,      40,      1;
  6:   40391,    65546,    35282,     8400,     960,     51,     1;
  7:  235416,   490687,   344421,   110321,   17960,   1520,    63,    1;
  8: 1372105,  3604679,  3210773,  1328910,  288961,  34629,  2275,   76,  1;
  9: 7997214, 26114171, 28946442, 15060642, 4200560, 668360, 61866, 3262, 90, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-86bd9191d575b368
• 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-86bd9191d575b368
• 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-86bd9191d575b368

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

• basis_map A008277 e-86bd9191d575b368

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle of coefficients of Chebyshev's S(n,x+5) 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-bc1afad92d84942a root: theorem check

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

    • theorem_check three_term_recurrence e-742fc3935df1d1d3 root: theorem check

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

    • preserver_image affine, shift 0 e-dbf8101c811237e4

      Affine, reversal and Möbius substitutions (elementary) 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 (-6, -2)
        • 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-785b355515669be0

      Geometric basis map (elementary) 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 (expanded above)
        • ↵ theorem_check three_term_recurrence e-ec629e395a3def0c (expanded above)
        • 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023