A078812 → D

OEIS matches ?

• A001871: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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:
   1:   0;
   2:   1;
   3:   4,    2;
   4:  10,   12,    3;
   5:  20,   42,   24,    4;
   6:  35,  112,  108,   40,    5;
   7:  56,  252,  360,  220,   60,    6;
   8:  84,  504,  990,  880,  390,   84,   7;
   9: 120,  924, 2376, 2860, 1820,  630, 112,   8;
  10: 165, 1584, 5148, 8008, 6825, 3360, 952, 144, 9;

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

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (n + 1)*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n + 2 + (n + 1)*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-1dbefba2139b076a
• Coefficients of one sign. Every nonzero row, n >= 1: 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-1dbefba2139b076a
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-1dbefba2139b076a
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-1dbefba2139b076a

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-1dbefba2139b076a

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

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-537ef91063de36e5 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-5aa1c8cf88fab58c root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023