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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 (list; table; graph; refs; listen; history; text; internal format)
 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 (SageMath) # Function DivisorTriangle is defined in A327029, returns T(0, 0) = 1. 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

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Last modified January 18 19:43 EST 2020. Contains 331029 sequences. (Running on oeis4.)