A033282 → D

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:  0;
  1:  2;
  2:  5,   10;
  3:  9,   42,    42;
  4: 14,  112,   252,    168;
  5: 20,  240,   900,   1320,    660;
  6: 27,  450,  2475,   5940,   6435,   2574;
  7: 35,  770,  5775,  20020,  35035,  30030,   10010;
  8: 44, 1232, 12012,  56056, 140140, 192192,  136136,  38896;
  9: 54, 1872, 22932, 137592, 458640, 891072, 1002456, 604656, 151164;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n + (2*n^2 + 6*n)*x)*v[i - 1] + (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n^2 + 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-72304849d48b5982
• 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-72304849d48b5982
• 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-72304849d48b5982
• 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-72304849d48b5982

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-72304849d48b5982

  Laguerre’s theorem on differential operators [SP14] 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 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-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 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 (expanded above)
            • ↵ preserver_image reversal, shift 0 e-14da04f2a2dce119 (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 (expanded above)
            • ↵ theorem_check three_term_recurrence e-5bc6f2a826329589 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-e5f3b61b305da328 (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)

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

• A145596 → x ↦ x/(x+1)
• A145596 → x ↦ x+1

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