A318772 → x ↦ (1/243)x → x ↦ x+1

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:     1;
  1:     3;
  2:     9;
  3:    27;
  4:    82,    1;
  5:   249,    6;
  6:   756,   27;
  7:  2295,  108;
  8:  6967,  407, 1;
  9: 21150, 1476, 9;

The row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) + 3*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9, P(3,x) = 27.

P[0, x_] := 1;
P[1, x_] := 3;
P[2, x_] := 9;
P[3, x_] := 27;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] + 3*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; v[3] = 9; v[4] = 27; for(n = 4, N, my(i = n + 1); v[i] = (1 + x)*v[i - 4] + 3*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-fc673ff7432a109d
• 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-dbbb5664ea9ec441, e-fc673ff7432a109d
• 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-dbbb5664ea9ec441, e-fc673ff7432a109d
• 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-dbbb5664ea9ec441

Proof chains

2 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-fc673ff7432a109d

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image affine, shift 0 e-e3a638b293d2df1c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers paper_theorem e-7b49ab4fc0b1a646 root: theorem check

          Parent-row provers of polytool polytool ?

• preserver_image affine, shift 0 e-dbbb5664ea9ec441

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ preserver_image affine, shift 0 e-e3a638b293d2df1c (expanded above)

Related sequences ?

No candidate equalities or related sequences.