A039683: Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

OEIS matches ?

• A039683: triangle, 45 terms agree, 9 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A330797: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS

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,         1;
  2:          8,        -6,          1;
  3:        -48,        44,        -12,        1;
  4:        384,      -400,        140,      -20,        1;
  5:      -3840,      4384,      -1800,      340,      -30,       1;
  6:      46080,    -56448,      25984,    -5880,      700,     -42,      1;
  7:    -645120,    836352,    -420224,   108304,   -15680,    1288,    -56,    1;
  8:   10321920, -14026752,    7559936, -2153088,   359184,  -36288,   2184,  -72,   1;
  9: -185794560, 262803456, -150105600, 46315520, -8618400, 1012368, -75600, 3480, -90, 1;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2*n - 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] = -(2*n - 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)

Triangle begins:
   1:          1;
   2:         -2,         1;
   3:          8,        -6,          1;
   4:        -48,        44,        -12,        1;
   5:        384,      -400,        140,      -20,        1;
   6:      -3840,      4384,      -1800,      340,      -30,       1;
   7:      46080,    -56448,      25984,    -5880,      700,     -42,      1;
   8:    -645120,    836352,    -420224,   108304,   -15680,    1288,    -56,    1;
   9:   10321920, -14026752,    7559936, -2153088,   359184,  -36288,   2184,  -72,   1;
  10: -185794560, 262803456, -150105600, 46315520, -8618400, 1012368, -75600, 3480, -90, 1;

The row polynomials satisfy P(n,x) = -(2*n-2-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 - 2 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(2*n - 2 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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 >= 0, is zero or real-rooted with all roots in [1, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-8b470702f19ec444
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) 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-8b470702f19ec444
• 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-8b470702f19ec444

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

• preserver_image affine, shift 0 e-8b470702f19ec444

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-59b849691f316c6f root: theorem check

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

    • parent_provers root_domain e-ef36674c01230a86 root: theorem check

      Parent-row provers of polytool polytool ?

Used in 2 proof steps

• e-769dd7c64d51b4a5 obreschkoff_combination → obreschkoff_combination of Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(…
• e-769dd7c64d51b4a5 obreschkoff_combination → obreschkoff_combination of Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(…

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