A114656 → basis A007318

OEIS matches ?

• A003645: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:
   1:      1;
   2:      3,       1;
   3:     11,       8,       1;
   4:     45,      51,      15,       1;
   5:    197,     304,     146,      24,      1;
   6:    903,    1765,    1190,     330,     35,      1;
   7:   4279,   10128,    8835,    3520,    645,     48,     1;
   8:  20793,   57799,   62013,   32515,   8715,   1141,    63,    1;
   9: 103049,  329024,  419524,  274064,  98014,  19040,  1876,   80,  1;
  10: 518859, 1871145, 2766060, 2169300, 977130, 256158, 37884, 2916, 99, 1;

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

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x] + (6*n - 3 + (2*n - 1)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 2] + (6*n - 3 + (2*n - 1)*x)*v[i - 1])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-38a7985c4c1aa2e3
• Coefficients of one sign. Every nonzero row, n >= 1: 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-38a7985c4c1aa2e3, e-f0928c2ecfeb4b08
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-38a7985c4c1aa2e3, e-f0928c2ecfeb4b08
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-38a7985c4c1aa2e3, e-f0928c2ecfeb4b08
• Real-rooted. Every row P_n, n >= 1, 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-f0928c2ecfeb4b08

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-38a7985c4c1aa2e3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-e6c0603bd7c28948 root: theorem check

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

• definition_equality e-21dd78e181b91ed1 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-f0928c2ecfeb4b08

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check three_term_recurrence e-e6c0603bd7c28948 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A114687 → x ↦ x/(x+1)
• A114655 → x ↦ x+1

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