A062264 → 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:   5;
  2:  12,   30;
  3:  21,  126,   105;
  4:  32,  336,   672,    280;
  5:  45,  720,  2520,   2520,    630;
  6:  60, 1350,  7200,  12600,   7560,   1260;
  7:  77, 2310, 17325,  46200,  48510,  19404,   2310;
  8:  96, 3696, 36960, 138600, 221760, 155232,  44352,  3960;
  9: 117, 5616, 72072, 360360, 810810, 864864, 432432, 92664, 6435;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 4*n + (3*n^2 + 6*n + 6)*x)*v[i - 1] + ((2*n + 3)*x - (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); 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-3c18fb3613f81561
• 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-3c18fb3613f81561
• 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-3c18fb3613f81561
• 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-3c18fb3613f81561

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-3c18fb3613f81561

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

  • premise (sequence): Coefficient triangle of certain polynomials N(4; m,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c22f7d753efa18e6 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.