OFFSET
1,2
COMMENTS
T(n,k) is the number of k-element subsets of {1,...,n} whose mean is an integer. Row sums and alternating row sums: A051293 and A000027. - Clark Kimberling, May 05 2012 [first link corrected to A051293 by Antti Karttunen, Feb 18 2013]
LINKS
FORMULA
T(n,k) = C(n,k) - Sum[a_1=1..(n-k+1)] Sum[a_2=a_1+1..(n-k+2)] ... Sum[a_k=a_(k-1)+1..n] (ceiling(f(a_1,...a_k)) - floor(f(a_1,...a_k))), where f(a_1,...a_k) = (a_1+...+a_k)/k is the arithmetic mean. - Ctibor O. Zizka, Jun 03 2015
EXAMPLE
The third term of the sixth row is 8 because we have solutions {1+2+3, 1+2+6, 1+3+5, 1+5+6, 2+3+4, 2+4+6, 3+4+5, 4+5+6} which all are divisible by 3.
From Clark Kimberling, May 05 2012: (Start)
First six rows:
1;
2, 0;
3, 1, 1;
4, 2, 2, 0;
5, 4, 4, 1, 1;
6, 6, 8, 4, 2, 0;
(End)
MAPLE
[seq(DivSumChooseTriangle(j), j=1..120)]; DivSumChooseTriangle := (n) -> nops(DivSumChoose(trinv(n-1), (n-((trinv(n-1)*(trinv(n-1)-1))/2))));
DIVSum_SOLUTIONS_GLOBAL := []; DivSumChoose := proc(n, k) global DIVSum_SOLUTIONS_GLOBAL; DIVSum_SOLUTIONS_GLOBAL := []; DivSumChooseSearch([], n, k); RETURN(DIVSum_SOLUTIONS_GLOBAL); end;
DivSumChooseSearch := proc(s, n, k) global DIVSum_SOLUTIONS_GLOBAL; local i, p; p := nops(s); if(p = k) then if(0 = (convert(s, `+`) mod k)) then DIVSum_SOLUTIONS_GLOBAL := [op(DIVSum_SOLUTIONS_GLOBAL), s]; fi; else for i from lmax(s)+1 to n-(k-p)+1 do DivSumChooseSearch([op(s), i], n, k); od; fi; end;
lmax := proc(a) local e, z; z := 0; for e in a do if whattype(e) = list then e := last_term(e); fi; if e > z then z := e; fi; od; RETURN(z); end;
# second Maple program:
b:= proc(n, s, m, t) option remember; `if`(n=0, `if`(s=0 and t=0, 1, 0),
`if`(t=0, 0, b(n-1, irem(s+n, m), m, t-1))+b(n-1, s, m, t))
end:
T:= (n, k)-> b(n, 0, k$2):
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Aug 28 2018
MATHEMATICA
t[n_, k_] := Length[ Select[ Subsets[ Range[n], {k}], Mod[Total[#], k] == 0 & ]]; Flatten[ Table[ t[n, k], {n, 1, 13}, {k, 1, n}]] (* Jean-François Alcover, Dec 02 2011 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 11 2001
EXTENSIONS
Starting offset corrected from 0 to 1 by Antti Karttunen, Feb 18 2013.
STATUS
approved