A161198 → basis A013610

OEIS matches ?

• A051581: 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:         3,          6;
   3:        15,         48,          36;
   4:       105,        426,         540,         216;
   5:       945,       4464,        7416,        5184,        1296;
   6:     10395,      54774,      108360,      101520,       45360,        7776;
   7:    135135,     774432,     1737324,     1969920,     1198800,      373248,      46656;
   8:   2027025,   12427290,    30706452,    39972744,    29801520,    12791520,    2939328,     279936;
   9:  34459425,  223426080,   596573424,   863775360,   746462304,   396264960,  126717696,   22394880,   1679616;
  10: 654729075, 4451852070, 12675451536, 19991172384, 19365435936, 12007808064, 4785225984, 1185808896, 166281984, 10077696;

The row polynomials satisfy P(n,x) = (2*n-1+6*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 1 + 6*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-1b98e7733a40a4e4
• 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-1b98e7733a40a4e4
• 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-1b98e7733a40a4e4

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-1b98e7733a40a4e4

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-6c4d7be2cbbf8954 root: theorem check

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

    • parent_provers root_domain e-84f434121cdb02d1 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