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

Author

R. J. Mathar, Jun 12 2019

Keywords

• nonn
• easy
• tabl

Examples

			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;
  ...
		

Links

• 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).

Crossrefs

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

Programs

Maple

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

Formula

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