A108838 → x ↦ x+1

OEIS matches ?

• A010683: 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:     2;
  1:     5,      2;
  2:    14,     12,      2;
  3:    42,     56,     21,      2;
  4:   132,    240,    144,     32,      2;
  5:   429,    990,    825,    300,     45,     2;
  6:  1430,   4004,   4290,   2200,    550,    60,     2;
  7:  4862,  16016,  21021,  14014,   5005,   924,    77,    2;
  8: 16796,  63648,  99008,  81536,  38220, 10192,  1456,   96,   2;
  9: 58786, 251940, 453492, 445536, 259896, 91728, 19110, 2184, 117, 2;

The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = -(2*n^2+n-3)*x^2*P(n-2,x) + (8*n^2+16*n+6+(4*n^2+8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 2*x.

P[0, x_] := 2;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + n - 3)*x^2*P[n - 2, x] + (8*n^2 + 16*n + 6 + (4*n^2 + 8*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 + n - 3)*x^2*v[i - 2] + (8*n^2 + 16*n + 6 + (4*n^2 + 8*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-5ed8e98bdf56db24
• 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-454321f2b34b0823, e-5ed8e98bdf56db24
• 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-454321f2b34b0823, e-5ed8e98bdf56db24
• 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-454321f2b34b0823, e-5ed8e98bdf56db24
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-454321f2b34b0823

Proof chains

3 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 A007318 e-5ed8e98bdf56db24

  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 ?

• definition_equality e-6e78dd0e7e5c1533 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-454321f2b34b0823

  Affine, reversal and Möbius substitutions (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 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A007318 → ⊙ C(n+1,k+1) (shift +2) → x ↦ x/(x+1)
• A126216 → D

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