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 A309122 Sum of the sizes of all subsets of [n] whose sum is divisible by n. 3
 1, 1, 6, 6, 20, 34, 70, 124, 270, 516, 1034, 2060, 4108, 8198, 16440, 32760, 65552, 131142, 262162, 524312, 1048740, 2097162, 4194326, 8388856, 16777300, 33554444, 67109418, 134217764, 268435484, 536872072, 1073741854, 2147483632, 4294969404, 8589934608 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The bivariate g.f. of array T(n,k) = A267632(n,k) is Sum_{n, k >= 1} T(n,k) * x^n * y^k = -x/(1 - x) - Sum_{s >= 1} (phi(s)/s) * log(1 - x^s + (-x*y)^s). Differentiating w.r.t. y and setting y = 1, we get the g.f. of a(n) = k * Sum_{1 <= k <= n} T(n,k) (see below). - Petros Hadjicostas, Jul 13 2019 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 FORMULA a(n) = Sum_{k=1..n} k * A267632(n,k). From Petros Hadjicostas, Jul 13 2019: (Start) G.f.: Sum_{s >= 1} phi(s) * (-x)^(s-1)/(1 - x^s + (-x)^s) = -Sum_{m >= 1} phi(2*m) * x^(2*m-1) + Sum_{m >= 0} phi(2*m+1) * x^(2*m)/(1 - 2*x^(2*m+1)). a(2*m + 1) = A053636(2*m + 1)/2 = (1/2) * Sum_{d|2*m+1} phi(d) * 2^((2*m+1)/d) for m >= 0. a(2*m) = -phi(2*m) +  A053636(2*m)/2 for m >= 1. (End) EXAMPLE a(5) = 20 = 0 + 1 + 2 + 2 + 3 + 3 + 4 + 5 = |{}| + |{5}| + |{1,4}| + |{2,3}| + |{1,4,5}| + |{2,3,5}| + |{1,2,3,4}| + |{1,2,3,4,5}|. MAPLE b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],       b(n-1, m, s) +(g-> g+[0, g[1]])(b(n-1, m, irem(s+n, m))))     end: a:= proc(n) option remember; forget(b); b(n\$2, 0)[2] end: seq(a(n), n=1..40); CROSSREFS Cf. A000010, A053636, A267632, A309128. Sequence in context: A255468 A246037 A045896 * A115046 A004983 A298936 Adjacent sequences:  A309119 A309120 A309121 * A309123 A309124 A309125 KEYWORD nonn AUTHOR Alois P. Heinz, Jul 13 2019 STATUS approved

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Last modified January 20 03:25 EST 2022. Contains 350467 sequences. (Running on oeis4.)