A367270 → basis A013610

OEIS matches ?

• A388060: row sums, 14 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:
  0:     1;
  1:     1;
  2:     3,      6;
  3:    10,     36,      27;
  4:    35,    180,     270,     108;
  5:   126,    840,    1890,    1620,     405;
  6:   462,   3780,   11340,   15120,    8505,    1458;
  7:  1716,  16632,   62370,  113400,  102060,   40824,    5103;
  8:  6435,  72072,  324324,  748440,  935550,  612360,  183708,  17496;
  9: 24310, 308880, 1621620, 4540536, 7297290, 6735960, 3367980, 787320, 59049;

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(18*n^2-45*n+18)*x^2*P(n-2,x) + (8*n^2-16*n+6+(12*n^2-24*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 45*n + 18)*x^2*P[n - 2, x] + (8*n^2 - 16*n + 6 + (12*n^2 - 24*n + 12)*x)*P[n - 1, x])/(2*n^2 - 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 - 45*n + 18)*x^2*v[i - 2] + (8*n^2 - 16*n + 6 + (12*n^2 - 24*n + 12)*x)*v[i - 1])/(2*n^2 - 3*n)); 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-59715d0ff282a908
• 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-59715d0ff282a908
• 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-59715d0ff282a908
• 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-59715d0ff282a908

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-59715d0ff282a908

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-2b43df6c63e34146 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