A152828 → 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: 0;
  3: 1;
  4: 1;
  5: 1;
  6: 2, 2;
  7: 2, 2;
  8: 2, 2;
  9: 3, 6, 3;

The row polynomials satisfy (n^3-12*n^2+47*n-60)*P(n,x) = (n^3-3*n^2-52*n+216+(n^3-3*n^2-52*n+216)*x)*P(n-3,x) - (27*n-189+(54*n-378)*x+(27*n-189)*x^2)*d/dx P(n-3,x) + (27+81*x+81*x^2+27*x^3)*d^2/dx^2 P(n-3,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 1, P(4,x) = 1, P(5,x) = 1.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[4, x_] := 1;
P[5, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 - 52*n + 216 + (n^3 - 3*n^2 - 52*n + 216)*x)*P[n - 3, x] - (27*n - 189 + (54*n - 378)*x + (27*n - 189)*x^2)*D[P[n - 3, x], x] + (27 + 81*x + 81*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 - 12*n^2 + 47*n - 60)]];
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] = 0; v[4] = 1; v[5] = 1; v[6] = 1; for(n = 6, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 - 52*n + 216 + (n^3 - 3*n^2 - 52*n + 216)*x)*v[i - 3] - (27*n - 189 + (54*n - 378)*x + (27*n - 189)*x^2)*deriv(v[i - 3], x) + (27 + 81*x + 81*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 - 12*n^2 + 47*n - 60)); 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-916a7875f7a82d69
• 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-916a7875f7a82d69
• 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-916a7875f7a82d69

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-916a7875f7a82d69

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

  • premise (sequence): Triangle read by rows, A007318 rows repeated three times.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-9670611e62c74d1a root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

• A152828 → basis A013609
• A152828: Triangle read by rows, A007318 rows repeated three times.

References

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