A188440 → 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: 2;
  3: 2;
  4: 4,  8;
  5: 4,  8;
  6: 6, 24, 24;
  7: 6, 24, 24;
  8: 8, 48, 96, 64;
  9: 8, 48, 96, 64;

The row polynomials satisfy (n-2)*P(n,x) = (3*n-8+(2*n^2-6*n)*x)*P(n-1,x) - (2+(2*n+4)*x+4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 8 + (2*n^2 - 6*n)*x)*P[n - 1, x] - (2 + (2*n + 4)*x + 4*n*x^2)*D[P[n - 1, x], x])/(n - 2)]];
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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 8 + (2*n^2 - 6*n)*x)*v[i - 1] - (2 + (2*n + 4)*x + 4*n*x^2)*deriv(v[i - 1], x))/(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-107b5f015c60ec70
• 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-107b5f015c60ec70
• 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-107b5f015c60ec70

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-107b5f015c60ec70

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

  • premise (sequence): Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-b186cc46021ee4e1 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (10)

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

• A274498 → D
• A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
• A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k…
• A274496 → basis A013611
• A103633 → x ↦ x/(x+1)
• A152198 → basis A013611
• A188440 → basis A013610
• A274498 → basis A013610
• A274496 → x ↦ x+1
• A152198 → x ↦ x+1

References

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