A027467 → basis A013610

OEIS matches ?

• A001029: 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:
  0:           1;
  1:          16,            3;
  2:         256,           96,           9;
  3:        4096,         2304,         432,          27;
  4:       65536,        49152,       13824,        1728,          81;
  5:     1048576,       983040,      368640,       69120,        6480,        243;
  6:    16777216,     18874368,     8847360,     2211840,      311040,      23328,       729;
  7:   268435456,    352321536,   198180864,    61931520,    11612160,    1306368,     81648,     2187;
  8:  4294967296,   6442450944,  4227858432,  1585446912,   371589120,   55738368,   5225472,   279936,   6561;
  9: 68719476736, 115964116992, 86973087744, 38050725888, 10701766656, 2006581248, 250822656, 20155392, 944784, 19683;

The row polynomials satisfy P(n,x) = (16+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(16 + 3*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; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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 - 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-d2b5f968b40c2621
• 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-d2b5f968b40c2621
• 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-d2b5f968b40c2621

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-d2b5f968b40c2621

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-14b8bc50df3e26b6 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-5234407c4fbd8679 root: theorem check

      Parent-row provers of polytool polytool ?

    • basis_map A007318 e-7eb5590197b773bf

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (14 + x)^n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-08b3ad2096e0d7df root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • parent_provers periodic_product e-7403f1e58efb6a95 root: theorem check

          Parent-row provers of polytool polytool ?

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