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);
MATHEMATICA
b[n_, m_, s_] := b[n, m, s] = If[n == 0, {If[s == 0, 1, 0], 0},
b[n-1, m, s] + Function[g, g + {0, g[[1]]}][b[n-1, m, Mod[s+n, m]]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 13 2019
STATUS
approved