A274498 → x ↦ x+1

OEIS matches ?

• A100747: 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 ?

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:     1;
  1:     3;
  2:     9,     6;
  3:    27,    18;
  4:    81,   108,    36;
  5:   243,   324,   108;
  6:   729,  1458,   972,   216;
  7:  2187,  4374,  2916,   648;
  8:  6561, 17496, 17496,  7776, 1296;
  9: 19683, 52488, 52488, 23328, 3888;

The row polynomials satisfy P(n,x) = (9+6*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.

P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(9 + 6*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (9 + 6*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-f41ab3c7deb8e5d8
• 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-94bbecf72edd8f15, e-f41ab3c7deb8e5d8
• 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-94bbecf72edd8f15, e-f41ab3c7deb8e5d8
• 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-94bbecf72edd8f15

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 A007318 e-f41ab3c7deb8e5d8

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-363ec3019ccc29a9 root: theorem check

      Parent-row provers of polytool polytool ?

• definition_equality e-98d5ec9577026c22 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-94bbecf72edd8f15

  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 degree of asymmetry equal to k (n>=0; 0…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-363ec3019ccc29a9 (expanded above)

Related sequences ?

Same fingerprint (3)

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

• A188440 → basis A007318
• A152198 → basis A013610
• A274496 → basis A013610