A008955 → basis A013609

OEIS matches ?

• A277353: 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:            2,             2;
  2:           10,            26,             16;
  3:          100,           440,            628,             288;
  4:         1700,         10680,          24756,           24992,            9216;
  5:        44200,        362680,        1177656,         1887592,         1489216,          460800;
  6:      1635400,      16601560,       69686232,       154632136,       191007616,       124273152,         33177600;
  7:     81770000,     990347200,     5111264480,     14560857536,     24704330128,     24932403968,      13837648896,      3251404800;
  8:   5315050000,   74839128000,   458996632800,   1600697593280,   3469571222928,   4782760514304,    4090794886144,   1982560370688,    416179814400;
  9: 435834100000, 6997846596000, 49761662625600, 205614657162560, 543817850391456, 954256900287264, 1110252383981056, 825278721951744, 355301524832256, 67421129932800;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + 1 + 2*n^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; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 1 + 2*n^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-8c95d99ebe6d3ebc
• 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-8c95d99ebe6d3ebc
• 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-8c95d99ebe6d3ebc
• 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-8c95d99ebe6d3ebc

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-8c95d99ebe6d3ebc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c0780e435a2392b0 root: theorem check

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

    • parent_provers paper_theorem e-b6645f2226a58079 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