A391533 → basis A013609

OEIS matches ?

• A309779: 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:    9,    12,      4;
  1:   18,    42,     32,      8;
  2:   36,   120,    148,     80,      16;
  3:   72,   312,    536,    456,     192,      32;
  4:  144,   768,   1696,   1984,    1296,     448,      64;
  5:  288,  1824,   4928,   7360,    6560,    3488,    1024,    128;
  6:  576,  4224,  13504,  24576,   27840,   20096,    9024,   2304,    256;
  7: 1152,  9600,  35456,  76160,  104832,   95872,   58240,  22656,   5120,    512;
  8: 2304, 21504,  90112, 223232,  361984,  401408,  308224, 161792,  55552,  11264,  1024;
  9: 4608, 47616, 223232, 626688, 1170432, 1526784, 1419264, 940032, 434688, 133632, 24576, 2048;

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 12*x + 4*x^2.

P[0, x_] := 9 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 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] = 9 + 12*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 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-8866dbf598c953fc
• 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-8866dbf598c953fc
• 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-8866dbf598c953fc

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-8866dbf598c953fc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check

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

    • parent_provers periodic_product e-5e5f630c1e2baa8a 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