A086645 → D+1

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:
   1:   1;
   2:   2,    1;
   3:   7,    8,     1;
   4:  16,   45,    18,      1;
   5:  29,  168,   154,     32,      1;
   6:  46,  465,   840,    390,     50,      1;
   7:  67, 1056,  3267,   2904,    825,     72,     1;
   8:  92, 2093, 10010,  15015,   8008,   1547,    98,    1;
   9: 121, 3760, 25844,  59488,  52910,  18928,  2660,  128,   1;
  10: 154, 6273, 58752, 193596, 262548, 155142, 39984, 4284, 162, 1;

The row polynomials satisfy (2*n^3-13*n^2+29*n-22+(4*n^2-21*n+26)*x+(2*n-4)*x^2)*P(n,x) = (2*n^3-9*n^2+14*n-8+(6*n^3-31*n^2+39*n-2)*x+(12*n^2-55*n+62)*x^2+(6*n-12)*x^3)*P(n-1,x) + ((4*n^2-12*n+6)*x-(4*n^2-20*n+24)*x^2-(8*n-22)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 9*n^2 + 14*n - 8 + (6*n^3 - 31*n^2 + 39*n - 2)*x + (12*n^2 - 55*n + 62)*x^2 + (6*n - 12)*x^3)*P[n - 1, x] + ((4*n^2 - 12*n + 6)*x - (4*n^2 - 20*n + 24)*x^2 - (8*n - 22)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 13*n^2 + 29*n - 22 + (4*n^2 - 21*n + 26)*x + (2*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = ((2*n^3 - 9*n^2 + 14*n - 8 + (6*n^3 - 31*n^2 + 39*n - 2)*x + (12*n^2 - 55*n + 62)*x^2 + (6*n - 12)*x^3)*v[i - 1] + ((4*n^2 - 12*n + 6)*x - (4*n^2 - 20*n + 24)*x^2 - (8*n - 22)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 13*n^2 + 29*n - 22 + (4*n^2 - 21*n + 26)*x + (2*n - 4)*x^2)); 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-792191c5a125b519
• 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-792191c5a125b519
• 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-792191c5a125b519
• 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-792191c5a125b519

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-792191c5a125b519

  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 ?

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