A126216 → 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:     3,      1;
  2:    10,      7,      1;
  3:    35,     39,     12,      1;
  4:   126,    196,     98,     18,      1;
  5:   462,    930,    660,    200,     25,     1;
  6:  1716,   4257,   3960,   1725,    360,    33,     1;
  7:  6435,  19019,  22022,  12705,   3850,   595,    42,    1;
  8: 24310,  83512, 116116,  84084,  34034,  7700,   924,   52,  1;
  9: 92378, 361998, 588744, 515424, 263718, 80262, 14196, 1368, 63, 1;

The row polynomials satisfy (n^3+3*n^2+2*n+(2*n^2+3*n-2)*x+(n+2)*x^2)*P(n,x) = (4*n^3+10*n^2+4*n+(3*n^3+13*n^2+n-2)*x+(6*n^2+4*n+2)*x^2+3*n*x^3)*P(n-1,x) - ((2*n^2+5*n+2)*x+(2*n^2+9*n+4)*x^2+(4*n+4)*x^3+2*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 + 10*n^2 + 4*n + (3*n^3 + 13*n^2 + n - 2)*x + (6*n^2 + 4*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] - ((2*n^2 + 5*n + 2)*x + (2*n^2 + 9*n + 4)*x^2 + (4*n + 4)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (2*n^2 + 3*n - 2)*x + (n + 2)*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 + 10*n^2 + 4*n + (3*n^3 + 13*n^2 + n - 2)*x + (6*n^2 + 4*n + 2)*x^2 + 3*n*x^3)*v[i - 1] - ((2*n^2 + 5*n + 2)*x + (2*n^2 + 9*n + 4)*x^2 + (4*n + 4)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (2*n^2 + 3*n - 2)*x + (n + 2)*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-49264023c59aa288
• 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-49264023c59aa288
• 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-49264023c59aa288
• 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-49264023c59aa288

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-49264023c59aa288

  Laguerre’s theorem on differential operators [SP14] 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. [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