A169888 Number of n-member subsets of 1..2n whose elements sum to a multiple of n.

1, 2, 2, 8, 18, 52, 152, 492, 1618, 5408, 18452, 64132, 225432, 800048, 2865228, 10341208, 37568338, 137270956, 504171584, 1860277044, 6892335668, 25631327688, 95640829924, 357975249028, 1343650267288, 5056424257552, 19073789328752, 72108867620204

Offset: 0

Author

N. J. A. Sloane, Jul 07 2010, based on a letter from Jean-Claude Babois

Keywords

• nonn

Comments

This is twice A145855 (for n>0), which is the main entry for this problem.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1669

Crossrefs

Cf. A061865, A119358, A145855, A318431, A318432, A318433.

Column k=2 of A304482.

Programs

Maple

with(combinat): t0:=[]; for n from 1 to 8 do ans:=0; t1:=choose(2*n,n); for i in t1 do s1:=add(i[j],j=1..n); if s1 mod n = 0 then ans:=ans+1; fi; od: t0:=[op(t0),ans]; od:

Mathematica

a[n_] := Sum[(-1)^(n+d)*EulerPhi[n/d]*Binomial[2d, d]/n, {d, Divisors[n]}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 22 2012, after T. D. Noe's program in A145855 *)

PARI

a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(n+d)*eulerphi(n/d)*binomial(2*d, d)/n)); \\ Altug Alkan, Aug 27 2018, after T. D. Noe at A145855

Formula

a(n) = A061865(2n,n). - Alois P. Heinz, Aug 28 2018

a(n) ~ 2^(2*n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 28 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 26 2018