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A308630
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Triangle T(n,k) read by rows: the sum of all smallest parts among all k-compositions of n.
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2
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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
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OFFSET
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1,2
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LINKS
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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).
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FORMULA
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T(n,k) = k*sum_{j=1..floor(n/k)} binomial(n-(j-1)*k-2, k-2).
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EXAMPLE
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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;
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MAPLE
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add(j*binomial(n-(j-1)*k-2, k-2), j=1..floor(n/k)) ;
%*k ;
end proc:
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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