A046716 → x ↦ x/(x+1)

OEIS matches ?

• A001339: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,  2;
   3: 1,  5,    5;
   4: 1,  9,   23,    16;
   5: 1, 14,   65,   116,     65;
   6: 1, 20,  145,   470,    669,    326;
   7: 1, 27,  280,  1415,   3634,   4429,    1957;
   8: 1, 35,  490,  3535,  14084,  30681,   33375,   13700;
   9: 1, 44,  798,  7756,  43939, 147532,  284066,  283072,  109601;
  10: 1, 54, 1230, 15456, 117579, 558642, 1646714, 2878284, 2673321, 986410;

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -((n-2)*x^2+(2*n-4)*x^3+(n-2)*x^4)*P(n-2,x) + (1+(n+2)*x+(2*n+1)*x^2+n*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n - 2)*x^2 + (2*n - 4)*x^3 + (n - 2)*x^4)*P[n - 2, x] + (1 + (n + 2)*x + (2*n + 1)*x^2 + n*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (-((n - 2)*x^2 + (2*n - 4)*x^3 + (n - 2)*x^4)*v[i - 2] + (1 + (n + 2)*x + (2*n + 1)*x^2 + n*x^3)*v[i - 1])/(1 + 2*x + 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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5da7cb2fae934977
• 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-5da7cb2fae934977
• 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-5da7cb2fae934977

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 mobius, shift 0 e-5da7cb2fae934977

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Coefficients of a special case of Poisson-Charlier polynomials.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-3fcbce34648f287c root: theorem check

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

    • preserver_image reversal, shift 0 e-1f661e973532853d

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-503a12a637bb094e root: theorem check

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

        • preserver_image reversal, shift 0 e-10ee31afdeccb0f4

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Coefficients of a special case of Poisson-Charlier polynomials.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-3fcbce34648f287c (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A094816 → x ↦ x+1

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023