A069466 → basis A013609

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:          1;
  1:          4,           4;
  2:         36,          72,          24;
  3:        400,        1200,         960,          160;
  4:       4900,       19600,       25200,        11200,         1120;
  5:      63504,      317520,      564480,       423360,       120960,         8064;
  6:     853776,     5122656,    11642400,     12418560,      6209280,      1241856,       59136;
  7:   11778624,    82450368,   228324096,    317116800,    230630400,     83026944,    12300288,      439296;
  8:  165636900,  1325095200,  4328644320,   7420533120,   7135128000,   3805401600,  1037836800,   118609920,    3294720;
  9: 2363904400, 21275139600, 80094643200, 163526563200, 196231875840, 140165625600, 57503846400, 12322252800, 1120204800, 24893440;

The row polynomials satisfy n^2*P(n,x) = -(64*n^2-128*n+48)*x^2*P(n-2,x) + (16*n^2-16*n+4+(16*n^2-16*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.

P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(64*n^2 - 128*n + 48)*x^2*P[n - 2, x] + (16*n^2 - 16*n + 4 + (16*n^2 - 16*n + 4)*x)*P[n - 1, x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(64*n^2 - 128*n + 48)*x^2*v[i - 2] + (16*n^2 - 16*n + 4 + (16*n^2 - 16*n + 4)*x)*v[i - 1])/(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-aca51a80a394c762
• 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-aca51a80a394c762
• 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-aca51a80a394c762
• 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-aca51a80a394c762

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 A013609 e-aca51a80a394c762

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-7dc194e46d1f26fe root: theorem check

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

Related sequences ?

Same fingerprint (1)

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

• A008459 → basis A013609

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