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