A207823 → (θ+1)²

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,       4;
  2:     15,      32,       9;
  3:     56,     184,     108,      16;
  4:    209,     928,     837,     256,      25;
  5:    780,    4364,    5328,    2496,     500,      36;
  6:   2911,   19648,   30294,   19200,    5875,     864,     49;
  7:  10864,   85872,  160056,  128160,   53000,   11880,   1372,    64;
  8:  40545,  367296,  803142,  777984,  406375,  122976,  21609,  2048,   81;
  9: 151316, 1545492, 3878928, 4411584, 2788100, 1065204, 252448, 36352, 2916, 100;

The row polynomials satisfy (1152*n+96*n*x-(240*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (2304*n+2304+(1008*n^2+2496*n+1488)*x-(480*n^3+528*n^2-256*n-304)*x^2-(24*n^4+284*n^3+464*n^2+212*n+8)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (1728+1728*x-(1008*n^2+1008*n-432)*x^2-(912*n^2+912*n+64)*x^3+(12*n^4+24*n^3-232*n^2-244*n-40)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+n^2)*x^6)*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[((2304*n + 2304 + (1008*n^2 + 2496*n + 1488)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (24*n^4 + 284*n^3 + 464*n^2 + 212*n + 8)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (1728 + 1728*x - (1008*n^2 + 1008*n - 432)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (12*n^4 + 24*n^3 - 232*n^2 - 244*n - 40)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(1152*n + 96*n*x - (240*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)]];
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] = ((2304*n + 2304 + (1008*n^2 + 2496*n + 1488)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (24*n^4 + 284*n^3 + 464*n^2 + 212*n + 8)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (1728 + 1728*x - (1008*n^2 + 1008*n - 432)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (12*n^4 + 24*n^3 - 232*n^2 - 244*n - 40)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(1152*n + 96*n*x - (240*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)); 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-9b17678c30419d79
• 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-9b17678c30419d79
• 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-9b17678c30419d79
• 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-9b17678c30419d79

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-9b17678c30419d79

  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