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A108458
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Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.
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3
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1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 15, 0, 1, 9, 22, 37, 52, 0, 1, 17, 52, 99, 151, 203, 0, 1, 33, 130, 283, 471, 674, 877, 0, 1, 65, 340, 855, 1561, 2386, 3263, 4140, 0, 1, 129, 922, 2707, 5451, 8930, 12867, 17007, 21147, 0, 1, 257, 2572, 8919, 19921, 35098, 53411, 73681, 94828, 115975
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OFFSET
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1,6
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COMMENTS
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Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.
Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - Augustine O. Munagi, Jul 16 2007
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LINKS
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FORMULA
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T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).
Double e.g.f.: exp(exp(x)*(exp(y)-1)).
U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - Vladeta Jovovic, Jul 12 2007
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EXAMPLE
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Triangle T(n,k) starts:
1;
0,1;
0,1,2;
0,1,3,5;
0,1,5,10,15;
T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.
The arrays U(n,k) starts:
1 0 0 0 0 ...
1 1 1 1 1 ...
2 3 5 9 17 ...
5 10 22 52 130 ...
15 37 99 283 855 ...
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CROSSREFS
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Row sums of T(n, k) yield A124496(n, 1).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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