A008955 → basis A013611

OEIS matches ?

No OEIS matches.

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,              4;
  2:           10,             52,              64;
  3:          100,            880,            2512,             2304;
  4:         1700,          21360,           99024,           199936,           147456;
  5:        44200,         725360,         4710624,         15100736,         23827456,          14745600;
  6:      1635400,       33203120,       278744928,       1237057088,       3056121856,        3976740864,        2123366400;
  7:     81770000,     1980694400,     20445057920,     116486860288,     395269282048,      797836926976,      885609529344,       416179814400;
  8:   5315050000,   149678256000,   1835986531200,   12805580746240,   55513139566848,   153048336457728,   261810872713216,    253767727448064,   106542032486400;
  9: 435834100000, 13995693192000, 199046650502400, 1644917257300480, 8701085606263296, 30536220809192448, 71056152574787584, 105635676409823232, 90957190357057536, 34519618525593600;

The row polynomials satisfy P(n,x) = (n^2+1+4*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 + 4*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 + 4*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-bf0617bcebb15ed7
• 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-bf0617bcebb15ed7
• 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-bf0617bcebb15ed7
• 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-bf0617bcebb15ed7

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 A013611 e-bf0617bcebb15ed7

  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