A086645 → (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:
   1: 1;
   2: 1,   4;
   3: 1,  24,     9;
   4: 1,  60,   135,     16;
   5: 1, 112,   630,    448,      25;
   6: 1, 180,  1890,   3360,    1125,      36;
   7: 1, 264,  4455,  14784,   12375,    2376,     49;
   8: 1, 364,  9009,  48048,   75075,   36036,   4459,     64;
   9: 1, 480, 16380, 128128,  321750,  288288,  89180,   7680,    81;
  10: 1, 612, 27540, 297024, 1093950, 1575288, 909636, 195840, 12393, 100;

The row polynomials satisfy (9*n-18-(22*n^3-111*n^2+182*n-96)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (9*n-18-(22*n^3-111*n^2+74*n+120)*x+(4*n^5-28*n^4-37*n^3+329*n^2-357*n+14)*x^2+(12*n^5-52*n^4+63*n^3-30*n+4)*x^3)*P(n-1,x) + (18*x-(44*n^2-68*n+14)*x^2+(8*n^4-16*n^3+42*n^2-54*n-6)*x^3-(8*n^4-16*n^3-2*n^2+14*n-2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 18 - (22*n^3 - 111*n^2 + 74*n + 120)*x + (4*n^5 - 28*n^4 - 37*n^3 + 329*n^2 - 357*n + 14)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*P[n - 1, x] + (18*x - (44*n^2 - 68*n + 14)*x^2 + (8*n^4 - 16*n^3 + 42*n^2 - 54*n - 6)*x^3 - (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*D[P[n - 1, x], x])/(9*n - 18 - (22*n^3 - 111*n^2 + 182*n - 96)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((9*n - 18 - (22*n^3 - 111*n^2 + 74*n + 120)*x + (4*n^5 - 28*n^4 - 37*n^3 + 329*n^2 - 357*n + 14)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*v[i - 1] + (18*x - (44*n^2 - 68*n + 14)*x^2 + (8*n^4 - 16*n^3 + 42*n^2 - 54*n - 6)*x^3 - (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*deriv(v[i - 1], x))/(9*n - 18 - (22*n^3 - 111*n^2 + 182*n - 96)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*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-1b0ecf2b3e1dc4df
• 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-1b0ecf2b3e1dc4df
• 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-1b0ecf2b3e1dc4df
• 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-1b0ecf2b3e1dc4df

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-1b0ecf2b3e1dc4df

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