A340995 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the Euler totient function phi; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 8, 9, 4, 1, 0, 2, 16, 19, 14, 5, 1, 0, 6, 20, 42, 36, 20, 6, 1, 0, 4, 36, 72, 89, 60, 27, 7, 1, 0, 6, 44, 134, 184, 165, 92, 35, 8, 1, 0, 4, 68, 210, 376, 391, 279, 133, 44, 9, 1, 0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1

Offset: 0

Author

Alois P. Heinz, Feb 01 2021

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  4,  8,   9,   4,   1;
  0,  2, 16,  19,  14,   5,   1;
  0,  6, 20,  42,  36,  20,   6,   1;
  0,  4, 36,  72,  89,  60,  27,   7,   1;
  0,  6, 44, 134, 184, 165,  92,  35,   8,  1;
  0,  4, 68, 210, 376, 391, 279, 133,  44,  9,  1;
  0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Columns k=0-2 give (offsets may differ): A000007, A000010, A065093.

Row sums give A159929.

T(2n,n) gives A340994.

Programs

Maple

T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[phi](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, EulerPhi[n]], With[{q = Quotient[k, 2]},
     Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)

Formula

T(n,k) = [x^n] (Sum_{j>=1} phi(j)*x^j)^k.