A086645 → D

OEIS matches ?

• A002697: row sums, 15 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:   6,    2;
   4:  15,   30,     3;
   5:  28,  140,    84,      4;
   6:  45,  420,   630,    180,      5;
   7:  66,  990,  2772,   1980,    330,      6;
   8:  91, 2002,  9009,  12012,   5005,    546,     7;
   9: 120, 3640, 24024,  51480,  40040,  10920,   840,    8;
  10: 153, 6120, 55692, 175032, 218790, 111384, 21420, 1224, 9;

The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (2*n^2-5*n+3+(6*n^2-23*n+17)*x)*P(n-1,x) + ((4*n-4)*x-(4*n-4)*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 - 5*n + 3 + (6*n^2 - 23*n + 17)*x)*P[n - 1, x] + ((4*n - 4)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];
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 - 5*n + 3 + (6*n^2 - 23*n + 17)*x)*v[i - 1] + ((4*n - 4)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); 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-13f8569d458f0f6e
• 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-13f8569d458f0f6e
• 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-13f8569d458f0f6e
• 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-13f8569d458f0f6e

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-13f8569d458f0f6e

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

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-1375f2f1a11f9809 root: theorem check

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

    • preserver_image reversal, shift 0 e-a5e3cd5a8b6d6a1a

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-1375f2f1a11f9809 (expanded above)

Related sequences ?

Same fingerprint (4)

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

• A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
• A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
• A103327 → reverse
• A091042 → reverse

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