A104684 → 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:      3,       1;
   3:     12,       8,       1;
   4:     50,      54,      15,       1;
   5:    210,     320,     150,      24,      1;
   6:    882,    1750,    1190,     330,     35,      1;
   7:   3696,    9072,    8190,    3360,    630,     48,     1;
   8:  15444,   45276,   51282,   28350,   7980,   1092,    63,    1;
   9:  64350,  219648,  300300,  210672,  80850,  16800,  1764,   80,  1;
  10: 267410, 1042470, 1673100, 1429428, 702702, 200970, 32340, 2700, 99, 1;

The row polynomials satisfy (n^3-3*n^2+2*n+(2*n^2-6*n+4)*x+(n-1)*x^2)*P(n,x) = (4*n^3-10*n^2-2*n+12+(3*n^3-n^2-24*n+28)*x+(6*n^2-15*n+9)*x^2+(3*n-5)*x^3)*P(n-1,x) - ((2*n^2-n-3)*x+(2*n^2+3*n-7)*x^2+(4*n-2)*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 10*n^2 - 2*n + 12 + (3*n^3 - n^2 - 24*n + 28)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n - 5)*x^3)*P[n - 1, x] - ((2*n^2 - n - 3)*x + (2*n^2 + 3*n - 7)*x^2 + (4*n - 2)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n + (2*n^2 - 6*n + 4)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 10*n^2 - 2*n + 12 + (3*n^3 - n^2 - 24*n + 28)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n - 5)*x^3)*v[i - 1] - ((2*n^2 - n - 3)*x + (2*n^2 + 3*n - 7)*x^2 + (4*n - 2)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n + (2*n^2 - 6*n + 4)*x + (n - 1)*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-62274b400b6ee467
• 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-62274b400b6ee467
• 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-62274b400b6ee467
• 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-62274b400b6ee467

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-62274b400b6ee467

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(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-74c20febd3423e2e root: theorem check

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

    • preserver_image reversal, shift 0 e-5047c71cb920acb6

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-18de01688f1c88b0 root: theorem check

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

        • theorem_check three_term_recurrence e-93f30859d3148f4f root: theorem check

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

        • preserver_image reversal, shift 0 e-00d47151fcd2720e

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(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-74c20febd3423e2e (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