A046716 → basis A008277

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:      2,        1;
   3:      5,        6,         1;
   4:     16,       37,        14,          1;
   5:     65,      280,       198,         34,          1;
   6:    326,     2621,      3260,       1100,        104,         1;
   7:   1957,    29454,     61431,      36010,       7179,       436,        1;
   8:  13700,   386737,   1301034,    1216061,     415954,     55321,     2400,       1;
   9: 109601,  5818320,  30705004,   43288868,   22827630,   5123636,   489096,   16108,      1;
  10: 986410, 98830105, 802851464, 1646425544, 1258515328, 424983762, 67393052, 4846668, 125718, 1;

The row polynomials satisfy P(n,x) = -(n-2+(3*n-6)*x+(n-2)*x^2)*P(n-2,x) - ((3*n-6)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (n+(n-1)*x)*P(n-1,x) + (n-1)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 2 + (3*n - 6)*x + (n - 2)*x^2)*P[n - 2, x] - ((3*n - 6)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (n + (n - 1)*x)*P[n - 1, x] + (n - 1)*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(n - 2 + (3*n - 6)*x + (n - 2)*x^2)*v[i - 2] - ((3*n - 6)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (n + (n - 1)*x)*v[i - 1] + (n - 1)*x*deriv(v[i - 1], 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-3d7ab7bd147ddfc0
• 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-3d7ab7bd147ddfc0
• 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-3d7ab7bd147ddfc0

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

• basis_map A008277 e-3d7ab7bd147ddfc0

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • 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 ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023