A103633 → D

OEIS matches ?

• A203150: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1, 2;
  3: 0, 2, 3;
  4: 0, 2, 6,  4;
  5: 0, 0, 3,  8,  5;
  6: 0, 0, 3, 12, 15,  6;
  7: 0, 0, 0,  4, 15, 18,  7;
  8: 0, 0, 0,  4, 20, 36, 28,  8;
  9: 0, 0, 0,  0,  5, 24, 42, 32, 9;

The row polynomials satisfy (n^2-n+(4*n^2-10*n+6)*x+(4*n^2-12*n+8)*x^2)*P(n,x) = -(n^3-4*n^2+3*n+(6*n^3-23*n^2+19*n)*x+(12*n^3-48*n^2+46*n)*x^2+(8*n^3-36*n^2+44*n-8)*x^3)*P(n-1,x) + ((2*n^2-2*n)*x+(10*n^2-14*n)*x^2+(16*n^2-28*n+4)*x^3+(8*n^2-16*n+4)*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[(-(n^3 - 4*n^2 + 3*n + (6*n^3 - 23*n^2 + 19*n)*x + (12*n^3 - 48*n^2 + 46*n)*x^2 + (8*n^3 - 36*n^2 + 44*n - 8)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n)*x + (10*n^2 - 14*n)*x^2 + (16*n^2 - 28*n + 4)*x^3 + (8*n^2 - 16*n + 4)*x^4)*D[P[n - 1, x], x])/(n^2 - n + (4*n^2 - 10*n + 6)*x + (4*n^2 - 12*n + 8)*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] = (-(n^3 - 4*n^2 + 3*n + (6*n^3 - 23*n^2 + 19*n)*x + (12*n^3 - 48*n^2 + 46*n)*x^2 + (8*n^3 - 36*n^2 + 44*n - 8)*x^3)*v[i - 1] + ((2*n^2 - 2*n)*x + (10*n^2 - 14*n)*x^2 + (16*n^2 - 28*n + 4)*x^3 + (8*n^2 - 16*n + 4)*x^4)*deriv(v[i - 1], x))/(n^2 - n + (4*n^2 - 10*n + 6)*x + (4*n^2 - 12*n + 8)*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-a139b2269602238f
• 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-a139b2269602238f
• 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-a139b2269602238f

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-a139b2269602238f

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

  • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check

      Parent-row provers of polytool polytool ?

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.