A126216 → basis A013610

OEIS matches ?

• A078009: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A127848: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,       3;
  2:     11,      21,        9;
  3:     45,     126,      108,       27;
  4:    197,     726,      936,      486,       81;
  5:    903,    4125,     7110,     5670,     2025,      243;
  6:   4279,   23319,    50355,    54405,    30375,     8019,      729;
  7:  20793,  131628,   341712,   466830,   357210,   149688,    30618,    2187;
  8: 103049,  743052,  2253888,  3727080,  3635604,  2109240,   694008,  113724,   6561;
  9: 518859, 4197879, 14570604, 28300104, 33618564, 25045524, 11512368, 3070548, 413343, 19683;

The row polynomials satisfy (n+2)*P(n,x) = -(n-1+(6*n-6)*x+(9*n-9)*x^2)*P(n-2,x) + (6*n+3+(6*n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*v[i - 2] + (6*n + 3 + (6*n + 3)*x)*v[i - 1])/(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-51c23bff4d2cb648
• 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-51c23bff4d2cb648
• 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-51c23bff4d2cb648
• 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-51c23bff4d2cb648

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 A013610 e-51c23bff4d2cb648

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-b83534d04d299e48 root: theorem check

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

    • basis_map A007318 e-5fe7e1b16eb84690

      Geometric basis map (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)
    • preserver_image reversal, shift 0 e-ba4d874222d3ab4e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • 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 (-inf, 0]
        • theorem_check three_term_recurrence e-7909c64e1764eda7 root: theorem check

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

        • 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 (expanded above)
            • ↵ preserver_image reversal, shift 0 e-14da04f2a2dce119 (expanded above)
        • preserver_image reversal, shift 0 e-3adfaf26de79e7ad

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-b83534d04d299e48 (expanded above)
            • ↵ basis_map A007318 e-5fe7e1b16eb84690 (expanded above)
            • preserver_image affine, shift 0 e-57f57afe27d03083

              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)
                • ↵ preserver_image reversal, shift 0 e-14da04f2a2dce119 (expanded above)
    • ↵ preserver_image affine, shift 0 e-57f57afe27d03083 (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