A102547 → x ↦ (1/27)x → 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:
  0:     1;
  1:     3;
  2:     9;
  3:    28,    1;
  4:    87,    6;
  5:   270,   27;
  6:   838,  111,   1;
  7:  2601,  432,   9;
  8:  8073, 1620,  54;
  9: 25057, 5920, 276, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-3,x) + x*d/dx P(n-3,x) + 3*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9.

P[0, x_] := 1;
P[1, x_] := 3;
P[2, x_] := 9;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x] + x*D[P[n - 3, x], x] + 3*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; v[3] = 9; for(n = 3, N, my(i = n + 1); v[i] = (1 + x)*v[i - 3] + x*deriv(v[i - 3], x) + 3*v[i - 1]); 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-f7cac2f9abe8b6c3
• 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-f7cac2f9abe8b6c3
• 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-f7cac2f9abe8b6c3

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-f7cac2f9abe8b6c3

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

  • premise (sequence): Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 0 e-b8d37b5fe37403cf

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image reversal, shift 0 e-9cfbf7a52f4ba322

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • preserver_image reversal, shift 0 e-0e9e334768ea0a35

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • parent_provers paper_theorem e-eef18680cd27a7c9 root: theorem check

                  Parent-row provers of polytool polytool ?

        • ↵ parent_provers paper_theorem e-eef18680cd27a7c9 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [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