A120910 → basis A013609

OEIS matches ?

• A005053: 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:     9,      6;
  2:    27,     36,     12;
  3:    81,    162,    108,      24;
  4:   243,    648,    648,     288,      48;
  5:   729,   2430,   3240,    2160,     720,     96;
  6:  2187,   8748,  14580,   12960,    6480,   1728,    192;
  7:  6561,  30618,  61236,   68040,   45360,  18144,   4032,    384;
  8: 19683, 104976, 244944,  326592,  272160, 145152,  48384,   9216,   768;
  9: 59049, 354294, 944784, 1469664, 1469664, 979776, 435456, 124416, 20736, 1536;

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

P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, 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 + 2*x)*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-618e7f7547136f14, e-c5b76738fa9c73f6
• 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-618e7f7547136f14, e-c5b76738fa9c73f6, e-c9b24916032e41a4
• 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-618e7f7547136f14, e-c5b76738fa9c73f6, e-c9b24916032e41a4
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-c9b24916032e41a4
• 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-c9b24916032e41a4

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A013609 e-618e7f7547136f14

  Geometric basis map (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 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 ?

• basis_map A007318 e-c5b76738fa9c73f6

  Geometric basis map (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)
    • ↵ preserver_image reversal, shift 0 e-98e1992c4c48b815 (expanded above)
    • ↵ parent_provers periodic_product e-f91a02f39edc297f (expanded above)
• preserver_image affine, shift 0 e-c9b24916032e41a4

  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)
    • ↵ preserver_image reversal, shift 0 e-98e1992c4c48b815 (expanded above)
    • ↵ parent_provers periodic_product e-f91a02f39edc297f (expanded above)

Related sequences ?

Same fingerprint (23)

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

• A356546 → basis A013610
• A038208 → basis A013610
• A038338 → basis A013610
• A038221 → basis A013610
• A196347 → basis A013610
• A178822 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A085881 → basis A013610
• A013620 → D
• A156992 → basis A013610
• A028326 → basis A013610
• A120910 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A038243 → basis A013611
• A085880 → basis A013610
• A038325 → basis A013610
• A038220 → D
• basis-map coefficients of A113214, shifted → basis A013610
• A178820 → basis A013610
• A135838 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A344565 → basis A013610
• A178821 → basis A013610

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