A207823 → θ+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:
  0:      1;
  1:      4,      2;
  2:     15,     16,       3;
  3:     56,     92,      36,       4;
  4:    209,    464,     279,      64,      5;
  5:    780,   2182,    1776,     624,    100,      6;
  6:   2911,   9824,   10098,    4800,   1175,    144,     7;
  7:  10864,  42936,   53352,   32040,  10600,   1980,   196,    8;
  8:  40545, 183648,  267714,  194496,  81275,  20496,  3087,  256,   9;
  9: 151316, 772746, 1292976, 1102896, 557620, 177534, 36064, 4544, 324, 10;

The row polynomials satisfy -(48*n+8*n*x-2*n^3*x^2)*P(n,x) = -(96*n+96+(12*n^2+64*n+52)*x-(4*n^3+4*n^2-8*n-8)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (144+144*x-(12*n^2+12*n-44)*x^2-(8*n^2+8*n-4)*x^3-(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(96*n + 96 + (12*n^2 + 64*n + 52)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (144 + 144*x - (12*n^2 + 12*n - 44)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-48*n - 8*n*x + 2*n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(96*n + 96 + (12*n^2 + 64*n + 52)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (144 + 144*x - (12*n^2 + 12*n - 44)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-48*n - 8*n*x + 2*n^3*x^2)); 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-dfb02fa127afb5fc
• 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-dfb02fa127afb5fc
• 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-dfb02fa127afb5fc
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-dfb02fa127afb5fc

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 euler_polynomial, shift 0 e-dfb02fa127afb5fc

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-720cd5df5ac0e3b3 root: theorem check

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

    • preserver_image affine, shift 0 e-56c0c02612a8191c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)
        • theorem_check three_term_recurrence e-c7df247a48c8552e root: theorem check

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

    • theorem_check three_term_recurrence e-ec629e395a3def0c root: theorem check

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

    • basis_map A007318 e-9c2f7b8109ebb5e1

      Geometric basis map (elementary) polytool ?

      • premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-af8f298ca8f41adf root: theorem check

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

        • ↵ theorem_check three_term_recurrence e-c7df247a48c8552e (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023