A116412 → 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:    1;
  2:    6,     2;
  3:   21,    18,     3;
  4:   60,    90,    36,     4;
  5:  156,   342,   234,    60,     5;
  6:  384,  1116,  1116,   480,    90,     6;
  7:  912,  3312,  4419,  2760,   855,   126,    7;
  8: 2112,  9216, 15480, 12900,  5760,  1386,  168,   8;
  9: 4800, 24480, 49752, 52332, 31095, 10710, 2100, 216, 9;

The row polynomials satisfy -(6*n^3-28*n^2+42*n-20-(3*n^3+2*n^2-15*n+10)*x+(2*n^2-2)*x^2)*P(n,x) = -(12*n^3-44*n^2+48*n-16+(12*n^3-88*n^2+108*n-32)*x-(20*n^2-48*n+4)*x^2+6*n*x^3)*P(n-1,x) + ((12*n^2-20*n+8)*x+(24*n^2-48*n+16)*x^2+(3*n^2-21*n+2)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^3 - 44*n^2 + 48*n - 16 + (12*n^3 - 88*n^2 + 108*n - 32)*x - (20*n^2 - 48*n + 4)*x^2 + 6*n*x^3)*P[n - 1, x] + ((12*n^2 - 20*n + 8)*x + (24*n^2 - 48*n + 16)*x^2 + (3*n^2 - 21*n + 2)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(-6*n^3 + 28*n^2 - 42*n + 20 + (3*n^3 + 2*n^2 - 15*n + 10)*x - (2*n^2 - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^3 - 44*n^2 + 48*n - 16 + (12*n^3 - 88*n^2 + 108*n - 32)*x - (20*n^2 - 48*n + 4)*x^2 + 6*n*x^3)*v[i - 1] + ((12*n^2 - 20*n + 8)*x + (24*n^2 - 48*n + 16)*x^2 + (3*n^2 - 21*n + 2)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(-6*n^3 + 28*n^2 - 42*n + 20 + (3*n^3 + 2*n^2 - 15*n + 10)*x - (2*n^2 - 2)*x^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-29ee48d67514cd97
• 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-29ee48d67514cd97
• 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-29ee48d67514cd97
• 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-29ee48d67514cd97

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-29ee48d67514cd97

  Laguerre’s theorem on differential operators [SP14] 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 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 (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.