A108838 → basis A013610

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:     2;
  1:     5,      6;
  2:    14,     36,      18;
  3:    42,    168,     189,       54;
  4:   132,    720,    1296,      864,      162;
  5:   429,   2970,    7425,     8100,     3645,      486;
  6:  1430,  12012,   38610,    59400,    44550,    14580,     1458;
  7:  4862,  48048,  189189,   378378,   405405,   224532,    56133,    4374;
  8: 16796, 190944,  891072,  2201472,  3095820,  2476656,  1061424,  209952,  13122;
  9: 58786, 755820, 4081428, 12029472, 21051576, 22289904, 13931190, 4776408, 767637, 39366;

The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = -(18*n^2+9*n-27)*x^2*P(n-2,x) + (8*n^2+16*n+6+(12*n^2+24*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 6*x.

P[0, x_] := 2;
P[1, x_] := 5 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 + 9*n - 27)*x^2*P[n - 2, x] + (8*n^2 + 16*n + 6 + (12*n^2 + 24*n)*x)*P[n - 1, x])/(2*n^2 + 7*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 + 9*n - 27)*x^2*v[i - 2] + (8*n^2 + 16*n + 6 + (12*n^2 + 24*n)*x)*v[i - 1])/(2*n^2 + 7*n + 3)); 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-7d811e606f00941d
• 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-7d811e606f00941d
• 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-7d811e606f00941d
• 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-7d811e606f00941d

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-7d811e606f00941d

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-a4549745e8cd4154 root: theorem check

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

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