A033282 → basis A008288

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:      3,       2;
  2:     11,      20,        5;
  3:     45,     142,       91,       14;
  4:    197,     896,     1022,      378,       42;
  5:    903,    5378,     9210,     5910,     1518,      132;
  6:   4279,   31524,    73419,    70422,    30657,     6006,      429;
  7:  20793,  182558,   542685,   711920,   457743,   149292,    23595,    1430;
  8: 103049, 1050368,  3816428,  6452732,  5649072,  2689830,   698126,   92378,   4862;
  9: 518859, 6022018, 25926628, 54163252, 61122672, 39358566, 14766778, 3174886, 361114, 16796;

The row polynomials satisfy (2*n^3+3*n^2-5*n-6)*P(n,x) = -(2*n^3-9*n^2+7*n+6)*P(n-4,x) + (16*n^3-48*n^2+20*n+24+(8*n^3-24*n^2+10*n+12)*x)*P(n-3,x) - (28*n^3-42*n^2-10*n+12-(16*n^3-24*n^2-4*n+6)*x)*P(n-2,x) + (16*n^3-28*n-12+(8*n^3-14*n-6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 11 + 20*x + 5*x^2, P(3,x) = 45 + 142*x + 91*x^2 + 14*x^3.

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 11 + 20*x + 5*x^2;
P[3, x_] := 45 + 142*x + 91*x^2 + 14*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 9*n^2 + 7*n + 6)*P[n - 4, x] + (16*n^3 - 48*n^2 + 20*n + 24 + (8*n^3 - 24*n^2 + 10*n + 12)*x)*P[n - 3, x] - (28*n^3 - 42*n^2 - 10*n + 12 - (16*n^3 - 24*n^2 - 4*n + 6)*x)*P[n - 2, x] + (16*n^3 - 28*n - 12 + (8*n^3 - 14*n - 6)*x)*P[n - 1, x])/(2*n^3 + 3*n^2 - 5*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 11 + 20*x + 5*x^2; v[4] = 45 + 142*x + 91*x^2 + 14*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(2*n^3 - 9*n^2 + 7*n + 6)*v[i - 4] + (16*n^3 - 48*n^2 + 20*n + 24 + (8*n^3 - 24*n^2 + 10*n + 12)*x)*v[i - 3] - (28*n^3 - 42*n^2 - 10*n + 12 - (16*n^3 - 24*n^2 - 4*n + 6)*x)*v[i - 2] + (16*n^3 - 28*n - 12 + (8*n^3 - 14*n - 6)*x)*v[i - 1])/(2*n^3 + 3*n^2 - 5*n - 6)); 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 [-1564556493/268435456, -46056243/268435456]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-a25a92238630af22
• 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-54689045adf560eb, e-a25a92238630af22
• 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-54689045adf560eb, e-a25a92238630af22
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-54689045adf560eb

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008288 e-54689045adf560eb

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    • theorem_check three_term_recurrence e-5bc6f2a826329589 root: theorem check

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

    • preserver_image mobius, shift 0 e-e5f3b61b305da328

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-d285019b29e04549 root: theorem check

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

        • preserver_image reversal, shift 0 e-14da04f2a2dce119

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-d285019b29e04549 (expanded above)
• basis_map A008288 e-a25a92238630af22

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    • ↵ theorem_check three_term_recurrence e-5bc6f2a826329589 (expanded above)
    • ↵ preserver_image mobius, shift 0 e-e5f3b61b305da328 (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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023