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 A304482 Number A(n,k) of n-element subsets of [k*n] whose elements sum to a multiple of n. Square array A(n,k) with n, k >= 0 read by antidiagonals. 8
 1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 1, 0, 1, 4, 6, 8, 0, 0, 1, 5, 12, 30, 18, 1, 0, 1, 6, 20, 76, 126, 52, 0, 0, 1, 7, 30, 155, 460, 603, 152, 1, 0, 1, 8, 42, 276, 1220, 3104, 3084, 492, 0, 0, 1, 9, 56, 448, 2670, 10630, 22404, 16614, 1618, 1, 0, 1, 10, 72, 680, 5138, 28506, 98900, 169152, 91998, 5408, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS When k=1 the only subset of [n] with n elements is [n] which sums to n(n+1)/2 and hence for n>0 and n even A(n,1) is zero and for n odd A(n,1) is one. LINKS Marko Riedel et al., Number of n-element subsets divisible by n FORMULA A(n,k) = (-1)^n * (1/n) * Sum_{d|n} C(k*d,d)*(-1)^d*phi(n/d), boundary values A(0,0) = 1, A(n, 0) = 0, A(0, k) = 1. EXAMPLE Square array A(n,k) begins:   1, 1,   1,     1,      1,      1,       1,        1, ...   0, 1,   2,     3,      4,      5,       6,        7, ...   0, 0,   2,     6,     12,     20,      30,       42, ...   0, 1,   8,    30,     76,    155,     276,      448, ...   0, 0,  18,   126,    460,   1220,    2670,     5138, ...   0, 1,  52,   603,   3104,  10630,   28506,    64932, ...   0, 0, 152,  3084,  22404,  98900,  324516,   874104, ...   0, 1, 492, 16614, 169152, 960650, 3854052, 12271518, ... MAPLE with(numtheory): A:= (n, k)-> `if`(n=0, 1, add(binomial(k*d, d)*(-1)^(n+d)*               phi(n/d), d in divisors(n))/n): seq(seq(A(n, d-n), n=0..d), d=0..11); MATHEMATICA A[n_, k_] : = (-1)^n (1/n) Sum[Binomial[k d, d] (-1)^d EulerPhi[n/d], {d, Divisors[n]}]; A[0, 0] = 1; A[_, 0] = 0; A[0, _] = 1; Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 23 2019 *) PROG (PARI) T(n, k)=if(n==0, 1, (-1)^n*sumdiv(n, d, binomial(k*d, d) * (-1)^d * eulerphi(n/d))/n) for(n=0, 7, for(k=0, 7, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018 CROSSREFS Cf. A000007, A169888, A318431, A318432, A318433, A318557. Main diagonal gives A318477. Sequence in context: A322279 A292861 A292133 * A309021 A307968 A242464 Adjacent sequences:  A304479 A304480 A304481 * A304483 A304484 A304485 KEYWORD nonn,tabl AUTHOR Marko Riedel, Aug 28 2018 STATUS approved

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Last modified December 9 00:32 EST 2019. Contains 329871 sequences. (Running on oeis4.)