A064861 → x ↦ x+1

OEIS matches ?

• A101553: 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:     6,      7,      2;
  1:    18,     33,     20,      4;
  2:    36,     84,     73,     28,      4;
  3:   108,    324,    387,    230,     68,      8;
  4:   216,    756,   1098,    847,    366,     84,      8;
  5:   648,   2700,   4806,   4737,   2792,    984,    192,     16;
  6:  1296,   6048,  12312,  14280,  10321,   4760,   1368,    224,    16;
  7:  3888,  20736,  49032,  67464,  59523,  34922,  13624,   3408,   496,    32;
  8:  7776,  45360, 118800, 183960, 186510, 129367,  62170,  20440,  4400,   560,   32;
  9: 23328, 151632, 447120, 789480, 927450, 761121, 445244, 185660, 54080, 10480, 1216, 64;

The row polynomials satisfy P(n,x) = (6+7*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 7*x + 2*x^2, P(1,x) = 18 + 33*x + 20*x^2 + 4*x^3.

P[0, x_] := 6 + 7*x + 2*x^2;
P[1, x_] := 18 + 33*x + 20*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[(6 + 7*x + 2*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 7*x + 2*x^2; v[2] = 18 + 33*x + 20*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 7*x + 2*x^2)*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-d4e342c4e7f76e08
• 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-4e787ca2d31dd33d, e-d4e342c4e7f76e08
• 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-4e787ca2d31dd33d, e-d4e342c4e7f76e08
• 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-4e787ca2d31dd33d

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-d4e342c4e7f76e08

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check

      Parent-row provers of polytool polytool ?

• definition_equality e-579b5cde2da46ab8 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-4e787ca2d31dd33d

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-bbffc3f85cdc52e2 (expanded above)

Related sequences ?

No candidate equalities or related sequences.