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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.
4
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
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).
Sequence in context: A089000 A253829 A107238 * A055830 A293109 A233530
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 22 2015
STATUS
approved