A120909 → x ↦ x/(x+1)

OEIS matches ?

• A259031: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A164346: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A092898: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A002001: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A110594: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,  9;
  2: 3, 18,  27;
  3: 3, 27,  81,   81;
  4: 3, 36, 162,  324,   243;
  5: 3, 45, 270,  810,  1215,   729;
  6: 3, 54, 405, 1620,  3645,  4374,   2187;
  7: 3, 63, 567, 2835,  8505, 15309,  15309,   6561;
  8: 3, 72, 756, 4536, 17010, 40824,  61236,  52488,  19683;
  9: 3, 81, 972, 6804, 30618, 91854, 183708, 236196, 177147, 59049;

The row polynomials satisfy (1+x)*P(n,x) = (1+4*x+3*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[((1 + 4*x + 3*x^2)*P[n - 1, x])/(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] = ((1 + 4*x + 3*x^2)*v[i - 1])/(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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-e61ce0e3d93770a8
• 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-e61ce0e3d93770a8
• 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-e61ce0e3d93770a8

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 mobius, shift 0 e-e61ce0e3d93770a8

  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 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 (16)

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

• basis-map coefficients of A026729 → basis A013610
• A027465 → D
• basis-map coefficients of A030528, shifted → basis A013610
• basis-map coefficients of A112227 → basis A013610
• A013610 → D
• A038243 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A027465: Cube of lower triangular normalized binomial matrix.
• basis-map coefficients of A021012 → basis A013610
• A038279 → basis A013610
• A013610 → basis A013611
• A120910 → basis A007318
• A136572 → basis A013610
• A130123 → basis A013610
• basis-map coefficients of A026729, shifted → 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