A120910 → 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:     3;
  1:     9,      3;
  2:    27,     21,       3;
  3:    81,    111,      36,       3;
  4:   243,    525,     291,      54,       3;
  5:   729,   2343,    1980,     615,      75,      3;
  6:  2187,  10101,   12243,    5670,    1140,     99,     3;
  7:  6561,  42591,   71316,   46263,   13650,   1932,   126,    3;
  8: 19683, 176925,  399171,  348894,  141813,  29106,  3066,  156,   3;
  9: 59049, 727383, 2172780, 2492535, 1341585, 374661, 56700, 4626, 189, 3;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = (3 + x)*v[i - 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 - 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-056f2aef56ba708a
• 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-056f2aef56ba708a
• 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-056f2aef56ba708a

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-056f2aef56ba708a

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

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-26e15e96a906c1df root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-3e9b67ef5f31843c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-b63d4680864933c7

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-f5018d5f4ba5a310 root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • preserver_image reversal, shift 0 e-98e1992c4c48b815

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-26e15e96a906c1df (expanded above)
            • ↵ parent_provers periodic_product e-3e9b67ef5f31843c (expanded above)
        • parent_provers periodic_product e-f91a02f39edc297f root: theorem check

          Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
• A143495 → reverse
• A038207 → basis A008277

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
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