A028338 → x ↦ (−1/2)x+1 (shift +2)

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:               32,             -12,               1;
  1:             -384,             176,             -24,             1;
  2:             6144,           -3200,             560,           -40,             1;
  3:          -122880,           70144,          -14400,          1360,           -60,           1;
  4:          2949120,        -1806336,          415744,        -47040,          2800,         -84,           1;
  5:        -82575360,        53526528,       -13447168,       1732864,       -125440,        5152,        -112,        1;
  6:       2642411520,     -1795424256,       483835904,     -68898816,       5746944,     -290304,        8736,     -144,        1;
  7:     -95126814720,     67277684736,    -19213516800,    2964193280,    -275788800,    16197888,     -604800,    13920,     -180,     1;
  8:    3805072588800,  -2786234204160,    835818356736, -137781248000,   13995745280,  -923704320,    40389888, -1161600,    21120,  -220,    1;
  9: -167423193907200, 126399377571840, -39562241900544, 6898193268736, -753594040320, 54638735360, -2700859392, 91500288, -2090880, 30800, -264, 1;

The row polynomials satisfy P(n,x) = -(4*n+8-x)*P(n-1,x) for n >= 1, with P(0,x) = 32 - 12*x + x^2.

P[0, x_] := 32 - 12*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(4*n + 8 - x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 32 - 12*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(4*n + 8 - 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 [2, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-41f01d4be10aeae3
• 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-41f01d4be10aeae3
• 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-41f01d4be10aeae3

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 2 e-41f01d4be10aeae3

  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 >= 2, 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-f961be0564a72cf6 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 ar…
• e-f961be0564a72cf6 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 ar…

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