A120910 → θ+1

OEIS matches ?

• A064017: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    6,     6;
  2:   12,    24,     9;
  3:   24,    72,    54,    12;
  4:   48,   192,   216,    96,    15;
  5:   96,   480,   720,   480,   150,    18;
  6:  192,  1152,  2160,  1920,   900,   216,    21;
  7:  384,  2688,  6048,  6720,  4200,  1512,   294,   24;
  8:  768,  6144, 16128, 21504, 16800,  8064,  2352,  384,  27;
  9: 1536, 13824, 41472, 64512, 60480, 36288, 14112, 3456, 486, 30;

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

P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (2*n + 4)*x + (n + 1)*x^2)*P[n - 1, x])/(2 + n*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] = ((4 + (2*n + 4)*x + (n + 1)*x^2)*v[i - 1])/(2 + n*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-217a3f649e225959
• 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-217a3f649e225959
• 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-217a3f649e225959
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-217a3f649e225959

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 euler_polynomial, shift 0 e-217a3f649e225959

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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 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 (1)

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

• A038207 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.