login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 02:56 EDT 2020. Contains 334613 sequences. (Running on oeis4.)