A066571 Number of sets of positive integers with arithmetic mean n.

1, 3, 9, 31, 117, 479, 2061, 9183, 42021, 196239, 931457, 4480531, 21793257, 107004891, 529656765, 2640160039, 13241371629, 66771501151, 338333343825, 1721768732423, 8796192611917, 45096680384635, 231945566136129, 1196461977291959, 6188390166782849

Offset: 1

Author

Amarnath Murthy, Dec 19 2001

Keywords

• nonn
• nice

Comments

From Franklin T. Adams-Watters, Sep 13 2011: (Start)

If we use nonnegative integers instead of positive integers, we get this sequence shifted left (i.e., with offset 0).

The largest number that can be included in set of positive integers with mean n is the triangular number n*(n+1)/2 = A000217(n).

All values are odd. Sets including n are paired with the same set with n removed, with exception of {n}, as the empty set has no average.

(End)

Examples

			a(2) = 3 as there are three sets viz. {2}, {1,3}, {1,2,3}, each of which has the arithmetic mean 2.
a(3) = 9: the nine sets are {3}, {1, 5}, {2, 4}, {1, 2, 6}, {1, 3, 5}, {2, 3, 4}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}.
		

Links

• Martin Fuller, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A066572, A000217.

Cf. A072701, A164283.

Programs

Haskell

a066571 n = f [1..] 1 n 0 where
   f (k:ks) l nl x
     | y > nl  = 0
     | y < nl  = f ks (l + 1) (nl + n) y + f ks l nl x
     | otherwise = if y `mod` l == 0 then 1 else 0
     where y = x + k
-- Reinhard Zumkeller, Feb 13 2013

Maple

g := k->expand(mul(1+t*x^i,i=1..k)); A066571 := proc(n) local k; add(coeff(coeff(g(n*k),t,k),x,n*k),k=1..2*n-1); end;

Mathematica

g[k_] := Expand[Product[1 + t*x^i, {i, 1, k}]]; a[n_] := Sum[Coefficient[ Coefficient[g[n*k], t, k], x, n*k], {k, 1, 2*n - 1}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Feb 10 2018, translated from Maple *)

Formula

Sum of coefficient of t^k x^(n*k) in Product_{i=1..n*k} (1+t*x^i) for k <= 2*n-1. - N. J. A. Sloane

From Martin Fuller, Sep 14 2023: (Start)

Constant term in formal Laurent series (Product_{i=1-n..n*(n-1)/2} (1+x^i)) - 1.

a(n) = (Sum_{i=0..n*(n-1)/2} A053632(n-1,i)*A000009(i))*2-1. (End)

Extensions

Corrected and extended by N. J. A. Sloane, Dec 19 2001

More terms from Naohiro Nomoto, Jun 19 2002

More terms from David Wasserman, Sep 10 2002

More terms from Martin Fuller, Sep 14 2023