A011973 → x ↦ x−1/4 → D+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:   7,    4;
  3:  20,   16;
  4:  45,   72,    16;
  5:  86,  272,    96;
  6: 147,  812,   528,    64;
  7: 232, 2016,  2432,   512;
  8: 345, 4368,  8928,  3328,  256;
  9: 490, 8544, 27072, 17920, 2560;

The row polynomials satisfy (2*n^3-6*n^2+7*n+(8*n^2-22*n)*x+8*n*x^2)*P(n,x) = (2*n^3+n+3+(8*n^3-24*n^2+14*n-26)*x+(32*n^2-80*n+64)*x^2+(32*n-32)*x^3)*P(n-1,x) + ((4*n^2-4*n+2)*x-(16*n^2-32*n+20)*x^2-(64*n-64)*x^3-64*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[((2*n^3 + n + 3 + (8*n^3 - 24*n^2 + 14*n - 26)*x + (32*n^2 - 80*n + 64)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 2)*x - (16*n^2 - 32*n + 20)*x^2 - (64*n - 64)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 7*n + (8*n^2 - 22*n)*x + 8*n*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] = ((2*n^3 + n + 3 + (8*n^3 - 24*n^2 + 14*n - 26)*x + (32*n^2 - 80*n + 64)*x^2 + (32*n - 32)*x^3)*v[i - 1] + ((4*n^2 - 4*n + 2)*x - (16*n^2 - 32*n + 20)*x^2 - (64*n - 64)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 7*n + (8*n^2 - 22*n)*x + 8*n*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-fe74f628d5055d01
• 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-fe74f628d5055d01
• 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-fe74f628d5055d01

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 derivative_polynomial, shift 0 e-fe74f628d5055d01

  Laguerre’s theorem on differential operators [SP14] 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023