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Search: id:A124323
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| A124323 |
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Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n). |
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+0
2
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| 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 6, 0, 1, 11, 20, 10, 10, 0, 1, 41, 66, 60, 20, 15, 0, 1, 162, 287, 231, 140, 35, 21, 0, 1, 715, 1296, 1148, 616, 280, 56, 28, 0, 1, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 0, 1, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 0
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k)=binom(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n)=A052889(n)=n*B(n-1), where B(q) are the Bell numbers (A000110).
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FORMULA
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T(n,k)=binom(n,k)*[(-1)^(n-k)+sum((-1)^(j-1)*B(n-k-j), j=1..n-k)], where B(q) are the Bell numbers (A000110). E.g.f.=G(t,z)=exp[exp(z)-1+(t-1)z)].
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EXAMPLE
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T(4,2)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3, and 1|2|34 (if we take {1,2,3,4} as our 4-set).
Triangle starts
1;
0,1;
1,0,1;
1,3,0,1;
4,4,6,0,1;
11,20,10,10,0,1;
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MAPLE
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G:=exp(exp(z)-1+(t-1)*z): Gser:=simplify(series(G, z=0, 14)): for n from 0 to 11 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000110, A000296, A052889, A124324.
Sequence in context: A039727 A137176 A143949 this_sequence A106683 A139601 A079520
Adjacent sequences: A124320 A124321 A124322 this_sequence A124324 A124325 A124326
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2006
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