A011973 → x ↦ x−1/4 → 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:  2;
  2:  3,   8;
  3:  4,  32;
  4:  5,  80,    48;
  5:  6, 160,   288;
  6:  7, 280,  1008,   256;
  7:  8, 448,  2688,  2048;
  8:  9, 672,  6048,  9216,  1280;
  9: 10, 960, 12096, 30720, 12800;

The row polynomials satisfy -(n-(2*n^3+2*n^2)*x)*P(n,x) = -(n+1-(2*n^3+4*n^2-6*n+8)*x-(8*n^3+16*n^2-8*n-16)*x^2)*P(n-1,x) - (2*x-(4*n^2+12*n+16)*x^2+(16*n^2+48*n+32)*x^3)*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[(-(n + 1 - (2*n^3 + 4*n^2 - 6*n + 8)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*P[n - 1, x] - (2*x - (4*n^2 + 12*n + 16)*x^2 + (16*n^2 + 48*n + 32)*x^3)*D[P[n - 1, x], x])/(-n + (2*n^3 + 2*n^2)*x)]];
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] = (-(n + 1 - (2*n^3 + 4*n^2 - 6*n + 8)*x - (8*n^3 + 16*n^2 - 8*n - 16)*x^2)*v[i - 1] - (2*x - (4*n^2 + 12*n + 16)*x^2 + (16*n^2 + 48*n + 32)*x^3)*deriv(v[i - 1], x))/(-n + (2*n^3 + 2*n^2)*x)); 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-a96f03eb4914ee6b
• 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-a96f03eb4914ee6b
• 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-a96f03eb4914ee6b

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-a96f03eb4914ee6b

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

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 0 e-e2e7a7b682b47947

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

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

        • preserver_image reversal, shift 0 e-75f779345b39f910

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (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