A249248 → basis A013610

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:      4,       9;
  2:     11,      39,       27;
  3:     38,     195,      279,       81;
  4:    131,     864,     1863,     1404,      243;
  5:    490,    3975,    11502,    13905,     6156,      729;
  6:   1897,   18219,    66114,   111267,    85212,    25029,     2187;
  7:   7714,   85689,   374301,   808164,   887193,   459270,    96957,     6561;
  8:  32501,  409710,  2092608,  5525712,  7969995,  6106104,  2267190,   363042,   19683;
  9: 142142, 2005233, 11728026, 36639918, 65633976, 67330683, 37600362, 10501974, 1325322, 59049;

The row polynomials satisfy P(n,x) = (n+1+(3*n+3)*x)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.

P[0, x_] := 1;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (3*n + 3)*x)*P[n - 2, x] + (2 + 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; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (3*n + 3)*x)*v[i - 2] + (2 + 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-f39d681898f1de2d
• 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-f39d681898f1de2d
• 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-f39d681898f1de2d
• 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-f39d681898f1de2d

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-f39d681898f1de2d

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-2dcae0cd92b4b84a root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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