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A061865 Triangle in which the k-th item in the n-th row (both starting from 1) is the number of ways in which we can add k distinct integers from 1 to n, in such a way that the sum is divisible by k. 7
1, 2, 0, 3, 1, 1, 4, 2, 2, 0, 5, 4, 4, 1, 1, 6, 6, 8, 4, 2, 0, 7, 9, 13, 9, 5, 1, 1, 8, 12, 20, 18, 12, 4, 2, 0, 9, 16, 30, 32, 26, 14, 6, 1, 1, 10, 20, 42, 54, 52, 34, 18, 6, 2, 0, 11, 25, 57, 84, 94, 76, 48, 21, 7, 1, 1, 12, 30, 76, 126, 160, 152, 114, 64, 26, 6, 2, 0, 13, 36, 98, 181, 259, 284, 246, 163, 81, 28, 8, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Rows n = 1..150, flattened

Index entries for sequences related to subset sums modulo m

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))). This sequence f(a_1,...a_k) = (a_1+...+a_k)/k 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

The second diagonal is given by C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) = A002620, the third diagonal by A061866. Cf. A061857.

T(2n,n) gives A169888.

Sequence in context: A047983 A070812 A308230 * A135818 A078804 A071465

Adjacent sequences:  A061862 A061863 A061864 * A061866 A061867 A061868

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

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Last modified November 30 17:12 EST 2021. Contains 349424 sequences. (Running on oeis4.)