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 A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals. 26
 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 20, 33, 24, 5, 0, 0, 6, 30, 72, 96, 40, 6, 0, 0, 7, 42, 135, 280, 255, 84, 7, 0, 0, 8, 56, 228, 660, 1040, 780, 140, 8, 0, 0, 9, 72, 357, 1344, 3145, 4200, 2205, 288, 9, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Dirichlet convolution of phi(n) and k^n. - Richard L. Ollerton, May 07 2021 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = Sum_{d|n} phi(d)*k^(n/d). A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258170(n,i). - Alois P. Heinz, May 22 2015 G.f. for column k: Sum_{n>=1} phi(n)*k*x^n/(1-k*x^n) for k >= 0. - Petros Hadjicostas, Nov 06 2017 From Richard L. Ollerton, May 07 2021: (Start) A(n,k) = Sum_{i=1..n} k^gcd(n,i). A(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)). A(n,k) = A075195(n,k)*n for n >= 1, k >= 1. (End) EXAMPLE Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, ... 0, 1, 2, 3, 4, 5, 6, ... 0, 2, 6, 12, 20, 30, 42, ... 0, 3, 12, 33, 72, 135, 228, ... 0, 4, 24, 96, 280, 660, 1344, ... 0, 5, 40, 255, 1040, 3145, 7800, ... 0, 6, 84, 780, 4200, 15810, 46956, ... MAPLE with(numtheory): A:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA a[_, 0] = a[0, _] = 0; a[n_, k_] := Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *) CROSSREFS Columns k=0..10 give A000004, A001477, A053635, A054610, A054611, A054612, A054613, A054614, A054615, A054616, A054617. Rows n=0..10 give A000004, A001477, A002378, A054602, A054603, A054604, A054605, A054606, A054607, A054608, A054609. Main diagonal gives A228640. Cf. A000010, A075195, A258170. Sequence in context: A343042 A343046 A271917 * A265080 A228275 A228250 Adjacent sequences: A185648 A185649 A185650 * A185652 A185653 A185654 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 29 2013 STATUS approved

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Last modified May 18 12:17 EDT 2024. Contains 372630 sequences. (Running on oeis4.)