A249251 → 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:  3;
  2:  6,  12;
  3:  9,  42,   36;
  4: 12,  90,  180,    96;
  5: 15, 156,  513,   624,   240;
  6: 18, 240, 1116,  2232,  1920,   576;
  7: 21, 342, 2070,  5892,  8280,  5472,  1344;
  8: 24, 462, 3456, 12900, 25800, 27648, 14784,  3072;
  9: 27, 600, 5355, 24876, 65415, 99504, 85680, 38400, 6912;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (4*n - 8)*x - 8*x^2)*v[i - 1] + (x + 8*x^2 + 4*x^3)*deriv(v[i - 1], x))/(-n + 1 + (2*n - 2)*x)); 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-653cf308b211b3ef
• 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-653cf308b211b3ef
• 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-653cf308b211b3ef
• 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-653cf308b211b3ef

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-653cf308b211b3ef

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

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    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-67b682ca6511d839 root: theorem check

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

    • preserver_image reversal, shift 0 e-c7b023a4533e2987

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
        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-ee40b91a84fdd5b7 root: theorem check

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

        • preserver_image reversal, shift 0 e-c36241f3b645d7b5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
            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-67b682ca6511d839 (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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.