A120909 → Veronese 2

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: 3;
  2: 3,  12;
  3: 3,  36;
  4: 3,  72,   48;
  5: 3, 120,  240;
  6: 3, 180,  720,   192;
  7: 3, 252, 1680,  1344;
  8: 3, 336, 3360,  5376,  768;
  9: 3, 432, 6048, 16128, 6912;

The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-4*x)*P(n-2,x) for n >= 2, with P(0,x) = 3, P(1,x) = 3.

P[0, x_] := 3;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - 4*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - 4*x)*v[i - 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-3c4f109236c7a212
• 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-3c4f109236c7a212
• 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-3c4f109236c7a212

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

• veronese_section r=2, j=0, shift 0 e-3c4f109236c7a212

  Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] 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 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 (expanded above)
            • parent_provers periodic_product e-f91a02f39edc297f root: theorem check

              Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-f91a02f39edc297f (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A013609 → Veronese 2
• A086645 → γ

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
4. [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