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

0,5

Links

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

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)

Example

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;

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 *)

Prog

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

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.

Sequence in context: A089000 A253829 A107238 * A055830 A293109 A233530

Adjacent sequences: A258167 A258168 A258169 * A258171 A258172 A258173

Keyword

nonn,tabl

Author

Alois P. Heinz, May 22 2015

Status

approved