A102365 → D

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:    0;
  1:    0;
  2:    1;
  3:    5,      2;
  4:   18,     30,       3;
  5:   58,    258,     111,       4;
  6:  179,   1754,    1938,     332,       5;
  7:  543,  10560,   24090,   10740,     885,      6;
  8: 1636,  59316,  247830,  225140,   50010,   2202,    7;
  9: 4916, 319484, 2268654, 3678460, 1675810, 208662, 5243, 8;

The row polynomials satisfy (n-2)*P(n,x) = -(3*n-3+(2*n^2-8*n+6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) - ((2*n-2)*x+(2*n^2-7*n+5)*x^2-(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (4*n-7+(2*n^2-8*n+8)*x)*P(n-1,x) + ((2*n-4)*x-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] - ((2*n - 2)*x + (2*n^2 - 7*n + 5)*x^2 - (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (4*n - 7 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] + ((2*n - 4)*x - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] - ((2*n - 2)*x + (2*n^2 - 7*n + 5)*x^2 - (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (4*n - 7 + (2*n^2 - 8*n + 8)*x)*v[i - 1] + ((2*n - 4)*x - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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-2922bfe17192604c
• 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-2922bfe17192604c
• 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-2922bfe17192604c
• 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-2922bfe17192604c

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 derivative_polynomial, shift 0 e-2922bfe17192604c

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-1ce099339e80292c root: theorem check

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

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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.