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

%I A308630 #11 Oct 15 2024 16:48:53
%S A308630 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,
%T A308630 20,36,68,100,90,42,8,9,20,54,108,175,210,147,56,9,10,30,72,160,290,
%U A308630 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
%N A308630 Triangle T(n,k) read by rows: the sum of all smallest parts among all k-compositions of n.
%H A308630 Knopfmacher, Arnold; Munagi, Augustine O.  <a href="https://doi.org/10.1007/978-3-642-30979-3_11">Smallest parts in compositions</a>, 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).
%F A308630 T(n,k) = k*sum_{j=1..floor(n/k)} binomial(n-(j-1)*k-2, k-2).
%e A308630 The triangle starts in row n=1 with columns 1<=k<=n as:
%e A308630    1;
%e A308630    2,  2;
%e A308630    3,  2,  3;
%e A308630    4,  6,  6,  4;
%e A308630    5,  6,  9, 12,  5;
%e A308630    6, 12, 18, 24, 20,  6;
%e A308630    7, 12, 27, 40, 50, 30,  7;
%e A308630    8, 20, 36, 68,100, 90, 42,  8;
%e A308630    9, 20, 54,108,175,210,147, 56,  9;
%e A308630   10, 30, 72,160,290,420,392,224, 72, 10;
%e A308630   ...
%p A308630 A308630 := proc(n,k)
%p A308630     add(j*binomial(n-(j-1)*k-2,k-2),j=1..floor(n/k)) ;
%p A308630     %*k ;
%p A308630 end proc:
%Y A308630 Cf. A097941 (number of smallest parts), A002378 (k=2), A144677 (column k=3 divided by 3), A097940 (row sums).
%K A308630 nonn,easy,tabl
%O A308630 1,2
%A A308630 _R. J. Mathar_, Jun 12 2019