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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.

%I A258170 #37 Mar 24 2020 12:37:37
%S A258170 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,
%T A258170 63,301,350,140,21,1,0,8,136,972,1702,1050,266,28,1,0,9,261,3027,7770,
%U A258170 6951,2646,462,36,1,0,10,530,9355,34115,42526,22827,5880,750,45,1
%N 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.
%H A258170 Alois P. Heinz, <a href="/A258170/b258170.txt">Rows n = 0..140, flattened</a>
%F A258170 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i).
%F A258170 From _Petros Hadjicostas_, Sep 07 2018: (Start)
%F A258170 Conjecture 1: T(n,k) = Stirling2(n,k) for k >= 1 and k <= n <= 2*k - 1.
%F A258170 Conjecture 2: T(n,k) = Stirling2(n,k) for k >= 2 and n prime >= 2.
%F A258170 Here, Stirling2(n,k) = A008277(n,k).
%F A258170 (End)
%e A258170 Triangle T(n,k) begins:
%e A258170   0;
%e A258170   0,  1;
%e A258170   0,  2,   1;
%e A258170   0,  3,   3,    1;
%e A258170   0,  4,   8,    6,     1;
%e A258170   0,  5,  15,   25,    10,     1;
%e A258170   0,  6,  36,   91,    65,    15,     1;
%e A258170   0,  7,  63,  301,   350,   140,    21,    1;
%e A258170   0,  8, 136,  972,  1702,  1050,   266,   28,   1;
%e A258170   0,  9, 261, 3027,  7770,  6951,  2646,  462,  36,  1;
%e A258170   0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1;
%p A258170 with(numtheory):
%p A258170 A:= proc(n, k) option remember;
%p A258170       add(phi(d)*k^(n/d), d=divisors(n))
%p A258170     end:
%p A258170 T:= (n, k)-> add((-1)^(k-i)*binomial(k, i)*A(n, i), i=0..k)/k!:
%p A258170 seq(seq(T(n, k), k=0..n), n=0..12);
%t A258170 A[n_, k_] := A[n, k] = DivisorSum[n, EulerPhi[#]*k^(n/#)&];
%t A258170 T[n_, k_] := Sum[(-1)^(k-i)*Binomial[k, i]*A[n, i], {i, 0, k}]/k!; T[0, 0] = 0;
%t A258170 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 25 2017, translated from Maple *)
%o A258170 (Sage) # uses[DivisorTriangle from A327029]
%o A258170 DivisorTriangle(euler_phi, stirling_number2, 10) # _Peter Luschny_, Aug 24 2019
%Y A258170 Columns k=0-1 give: A000004, A000027.
%Y A258170 Row sums give A258171.
%Y A258170 Main diagonal gives A057427.
%Y A258170 T(2*n+1,n+1) gives A129506(n+1).
%Y A258170 Cf. A008277, A185651, A327029.
%K A258170 nonn,tabl
%O A258170 0,5
%A A258170 _Alois P. Heinz_, May 22 2015