A091042 → 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:
   1: 1;
   2: 1,   6;
   3: 1,  20,    15;
   4: 1,  42,   105,     28;
   5: 1,  72,   378,    336,     45;
   6: 1, 110,   990,   1848,    825,     66;
   7: 1, 156,  2145,   6864,   6435,   1716,     91;
   8: 1, 210,  4095,  20020,  32175,  18018,   3185,   120;
   9: 1, 272,  7140,  49504, 121550, 116688,  43316,  5440,  153;
  10: 1, 342, 11628, 108528, 377910, 554268, 352716, 93024, 8721, 190;

The row polynomials satisfy -(6*n-9-(4*n^3-12*n^2+11*n-3)*x)*P(n,x) = -(6*n-9-(4*n^3-12*n^2-25*n+51)*x-(12*n^3-20*n^2-5*n+12)*x^2)*P(n-1,x) - (12*x-(8*n^2+6)*x^2+(8*n^2-6)*x^3)*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[(-(6*n - 9 - (4*n^3 - 12*n^2 - 25*n + 51)*x - (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*P[n - 1, x] - (12*x - (8*n^2 + 6)*x^2 + (8*n^2 - 6)*x^3)*D[P[n - 1, x], x])/(-6*n + 9 + (4*n^3 - 12*n^2 + 11*n - 3)*x)]];
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] = (-(6*n - 9 - (4*n^3 - 12*n^2 - 25*n + 51)*x - (12*n^3 - 20*n^2 - 5*n + 12)*x^2)*v[i - 1] - (12*x - (8*n^2 + 6)*x^2 + (8*n^2 - 6)*x^3)*deriv(v[i - 1], x))/(-6*n + 9 + (4*n^3 - 12*n^2 + 11*n - 3)*x)); 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-1081279861cd2a9c
• 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-1081279861cd2a9c
• 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-1081279861cd2a9c
• 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-1081279861cd2a9c

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-1081279861cd2a9c

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

  • premise (sequence): Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-1c0df90858517049 root: theorem check

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

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