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

0,8

Links

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

Formula

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

Example

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;
  ...

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 *)

Crossrefs

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

Row sums give A159929.

T(2n,n) gives A340994.

Sequence in context: A060086 A308680 A177975 * A363733 A062135 A190182

Adjacent sequences: A340992 A340993 A340994 * A340996 A340997 A340998

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 01 2021

Status

approved