A054619 Triangle T(n,k) = Sum_{d|k} phi(d)*n^(k/d).

1, 2, 6, 3, 12, 33, 4, 20, 72, 280, 5, 30, 135, 660, 3145, 6, 42, 228, 1344, 7800, 46956, 7, 56, 357, 2464, 16835, 118104, 823585, 8, 72, 528, 4176, 32800, 262800, 2097200, 16781472, 9, 90, 747, 6660, 59085, 532350, 4783023, 43053480, 387422001

Offset: 1

Author

N. J. A. Sloane, Apr 16 2000

Keywords

• nonn
• tabl

Examples

			1;
2, 6;
3, 12, 33;
4, 20, 72,  280;
5, 30, 135, 660,  3145;
6, 42, 228, 1344, 7800, 46956;
...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A054618, A054630, A054631, A185651 (transpose).

Main diagonal gives: A228640.

Programs

Maple

with(numtheory):
T:= (n, k)-> add(phi(d)*n^(k/d), d=divisors(k)):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 28 2013

Mathematica

T[n_, k_] := Sum[EulerPhi[d]*n^(k/d), {d, Divisors[k]}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 25 2015 *)

PARI

T(n, k) = sumdiv(k, d, eulerphi(d)*n^(k/d)); \\ Michel Marcus, Feb 25 2015