A122542 → basis A013610

OEIS matches ?

• A122690: 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:    1,     3;
   3:    3,    12,      9;
   4:    7,    39,     63,     27;
   5:   17,   120,    288,    270,     81;
   6:   41,   351,   1116,   1620,   1053,     243;
   7:   99,   996,   3933,   7722,   7857,    3888,     729;
   8:  239,  2763,  13023,  32211,  44793,   34749,   13851,   2187;
   9:  577,  7536,  41256, 123012, 217242,  231336,  144342,  48114,   6561;
  10: 1393, 20283, 126432, 441072, 944946, 1283526, 1100790, 572994, 164025, 19683;

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

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

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = (1 + 3*x)*v[i - 2] + (2 + 3*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, 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-7f7d545e75881e09
• 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-7f7d545e75881e09
• 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-7f7d545e75881e09
• 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-7f7d545e75881e09

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-7f7d545e75881e09

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-e90fe0b3eba80839 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003