A110441 → basis A013609

OEIS matches ?

• A154244: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:     4,      2;
  2:    14,     16,       4;
  3:    48,     88,      48,       8;
  4:   164,    416,     360,     128,      16;
  5:   560,   1816,    2176,    1216,     320,     32;
  6:  1912,   7552,   11616,    8960,    3680,    768,     64;
  7:  6528,  30400,   57216,   56640,   32000,  10368,   1792,   128;
  8: 22288, 119552,  266432,  323072,  233920, 103936,  27776,  4096,  256;
  9: 76096, 461984, 1190400, 1711872, 1517824, 862848, 315392, 71680, 9216, 512;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (4 + 2*x)*v[i - 1]); 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-94a352ca3242bd54
• 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-94a352ca3242bd54
• 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-94a352ca3242bd54
• 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-94a352ca3242bd54

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-94a352ca3242bd54

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangular array formed by the Mersenne numbers.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-c01294f1353c0463 root: theorem check

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

    • theorem_check three_term_recurrence e-02d1e2cf79c791bd 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