A121574 → basis A007318

OEIS matches ?

• A090017: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:     3,      1;
  2:    10,      7,      1;
  3:    33,     35,     11,      1;
  4:   109,    155,     76,     15,     1;
  5:   360,    642,    429,    133,    19,     1;
  6:  1189,   2550,   2160,    919,   206,    23,    1;
  7:  3927,   9841,  10101,   5479,  1689,   295,   27,   1;
  8: 12970,  37189,  44854,  29617, 11671,  2803,  400,  31,  1;
  9: 42837, 138305, 191693, 149285, 71798, 22064, 4325, 521, 35, 1;

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

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + (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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 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-1b44abb141022866
• 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-1b44abb141022866, e-85c02a915efd5463
• 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-1b44abb141022866, e-85c02a915efd5463
• 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-1b44abb141022866, e-85c02a915efd5463
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-85c02a915efd5463

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-1b44abb141022866

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
    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-81aa6611c032ec70 root: theorem check

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

• definition_equality e-9c4f08bbb123003e root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-85c02a915efd5463

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
    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-81aa6611c032ec70 (expanded above)

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