A308630 Triangle T(n,k) read by rows: the sum of all smallest parts among all k-compositions of n.

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 6, 9, 12, 5, 6, 12, 18, 24, 20, 6, 7, 12, 27, 40, 50, 30, 7, 8, 20, 36, 68, 100, 90, 42, 8, 9, 20, 54, 108, 175, 210, 147, 56, 9, 10, 30, 72, 160, 290, 420, 392, 224, 72, 10, 11, 30, 90, 224, 460, 756, 882, 672, 324, 90, 11, 12, 42, 120, 312, 700, 1272, 1764, 1680, 1080, 450, 110

Offset

1,2

Links

Table of n, a(n) for n=1..77.

Knopfmacher, Arnold; Munagi, Augustine O. Smallest parts in compositions, Kotsireas, Ilias S. (ed.) et al., Advances in combinatorics. 3rd Waterloo workshop on computer algebra (WWCA, W80) 2011, Waterloo, Canada, May 26-29, 2011. Berlin: Springer. 197-207 (2013).

Formula

T(n,k) = k*sum_{j=1..floor(n/k)} binomial(n-(j-1)*k-2, k-2).

Example

The triangle starts in row n=1 with columns 1<=k<=n as:
   1;
   2,  2;
   3,  2,  3;
   4,  6,  6,  4;
   5,  6,  9, 12,  5;
   6, 12, 18, 24, 20,  6;
   7, 12, 27, 40, 50, 30,  7;
   8, 20, 36, 68,100, 90, 42,  8;
   9, 20, 54,108,175,210,147, 56,  9;
  10, 30, 72,160,290,420,392,224, 72, 10;
  ...

Maple

A308630 := proc(n, k)
    add(j*binomial(n-(j-1)*k-2, k-2), j=1..floor(n/k)) ;
    %*k ;
end proc:

Crossrefs

Cf. A097941 (number of smallest parts), A002378 (k=2), A144677 (column k=3 divided by 3), A097940 (row sums).

Sequence in context: A091256 A003990 A287958 * A059896 A079542 A220370

Adjacent sequences: A308627 A308628 A308629 * A308631 A308632 A308633

Keyword

nonn,easy,tabl

Author

R. J. Mathar, Jun 12 2019

Status

approved