A258170 T(n,k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i); triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

0, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 8, 6, 1, 0, 5, 15, 25, 10, 1, 0, 6, 36, 91, 65, 15, 1, 0, 7, 63, 301, 350, 140, 21, 1, 0, 8, 136, 972, 1702, 1050, 266, 28, 1, 0, 9, 261, 3027, 7770, 6951, 2646, 462, 36, 1, 0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1

Offset: 0

Author

Alois P. Heinz, May 22 2015

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  2,   1;
  0,  3,   3,    1;
  0,  4,   8,    6,     1;
  0,  5,  15,   25,    10,     1;
  0,  6,  36,   91,    65,    15,     1;
  0,  7,  63,  301,   350,   140,    21,    1;
  0,  8, 136,  972,  1702,  1050,   266,   28,   1;
  0,  9, 261, 3027,  7770,  6951,  2646,  462,  36,  1;
  0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-1 give: A000004, A000027.

Row sums give A258171.

Main diagonal gives A057427.

T(2*n+1,n+1) gives A129506(n+1).

Cf. A008277, A185651, A327029.

Programs

Maple

with(numtheory):
A:= proc(n, k) option remember;
      add(phi(d)*k^(n/d), d=divisors(n))
    end:
T:= (n, k)-> add((-1)^(k-i)*binomial(k, i)*A(n, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

A[n_, k_] := A[n, k] = DivisorSum[n, EulerPhi[#]*k^(n/#)&];
T[n_, k_] := Sum[(-1)^(k-i)*Binomial[k, i]*A[n, i], {i, 0, k}]/k!; T[0, 0] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 25 2017, translated from Maple *)

Sage

# uses[DivisorTriangle from A327029]
DivisorTriangle(euler_phi, stirling_number2, 10) # Peter Luschny, Aug 24 2019

Formula

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i).

From Petros Hadjicostas, Sep 07 2018: (Start)

Conjecture 1: T(n,k) = Stirling2(n,k) for k >= 1 and k <= n <= 2*k - 1.

Conjecture 2: T(n,k) = Stirling2(n,k) for k >= 2 and n prime >= 2.

Here, Stirling2(n,k) = A008277(n,k).

(End)