A133087 → 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:  2,   2;
  3:  5,   8,   3;
  4:  4,  12,  12,   4;
  5:  9,  32,  42,  24,   5;
  6:  6,  30,  60,  60,  30,   6;
  7: 13,  72, 165, 200, 135,  48,   7;
  8:  8,  56, 168, 280, 280, 168,  56,  8;
  9: 17, 128, 420, 784, 910, 672, 308, 80, 9;

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(3*n^3-15*n^2+22*n-8+(2*n^3-9*n^2+11*n-2)*x)*P(n-1,x) + (3*n^2-7*n+3+(5*n^2-11*n+4)*x+(2*n^2-4*n+1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2 + 2*x.

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 15*n^2 + 22*n - 8 + (2*n^3 - 9*n^2 + 11*n - 2)*x)*P[n - 1, x] + (3*n^2 - 7*n + 3 + (5*n^2 - 11*n + 4)*x + (2*n^2 - 4*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 2 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n^3 - 15*n^2 + 22*n - 8 + (2*n^3 - 9*n^2 + 11*n - 2)*x)*v[i - 1] + (3*n^2 - 7*n + 3 + (5*n^2 - 11*n + 4)*x + (2*n^2 - 4*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 3*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-14b2653a33368494
• 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-14b2653a33368494
• 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-14b2653a33368494

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-14b2653a33368494

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

  • premise (sequence): A133080 * A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A007318 e-3537980ff71b1a89

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-aa3a6b002de14e0c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-28a1cf3333f63fa4

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-4d02ac33f5b2fbc9 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

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