A122431 → D

OEIS matches ?

• A136725: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A181389: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A281813: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A086570: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A017113: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 3, 6, 3;
  1: 1, 6, 9,  4;
  2: 0, 2, 9, 12,  5;
  3: 0, 0, 3, 12, 15,  6;
  4: 0, 0, 0,  4, 15, 18,  7;
  5: 0, 0, 0,  0,  5, 18, 21,  8;
  6: 0, 0, 0,  0,  0,  6, 21, 24,  9;
  7: 0, 0, 0,  0,  0,  0,  7, 24, 27, 10;
  8: 0, 0, 0,  0,  0,  0,  0,  8, 27, 30, 11;
  9: 0, 0, 0,  0,  0,  0,  0,  0,  9, 30, 33, 12;

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

P[0, x_] := 3 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n*x + (n + 3)*x^2)*P[n - 1, x])/(n - 1 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + 6*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n*x + (n + 3)*x^2)*v[i - 1])/(n - 1 + (n + 2)*x)); 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-b524f7ca945d9ab3
• 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-b524f7ca945d9ab3
• 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-b524f7ca945d9ab3

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-b524f7ca945d9ab3

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

  • premise (sequence): Riordan array ((1+x)^3,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-73f8c6c9e8d957af 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.