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Search: id:A056242
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| A056242 |
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Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1<=k<=n). |
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+0
5
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| 1, 1, 2, 1, 5, 4, 1, 9, 16, 8, 1, 14, 41, 44, 16, 1, 20, 85, 146, 112, 32, 1, 27, 155, 377, 456, 272, 64, 1, 35, 259, 833, 1408, 1312, 640, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512, 1, 65, 870, 5202
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Generalized Riordan array (1/(1-x), x/(1-x)+x*dif(x/1-x),x)). - Paul Barry (pbarry(AT)wit.ie), Dec 26 2007
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REFERENCES
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Hwang, F. K.; Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
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FORMULA
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Reference gives explicit formula.
T(n, k)=sum((-1)^(k-1-j)*binomial(k-1, j)*binomial(n+2j-1, 2j), j=0..k-1) (1<=k<=n); this is formula (11) in the reference.
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EXAMPLE
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1; 1,2; 1,5,4; 1,9,16,8; 1,14,41,44,16; ...
T(3,2)=5 because we have {1}{23}, {23}{1}, {12}{3}, {3]{12} and {2}{13}.
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MAPLE
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T:=proc(n, k) if k=1 then 1 elif k<=n then sum((-1)^(k-1-j)*binomial(k-1, j)*binomial(n+2*j-1, 2*j), j=0..k-1) else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..12);
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CROSSREFS
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Second diagonal gives A053220.
Sequence in context: A113178 A108362 A141506 this_sequence A128718 A112358 A126351
Adjacent sequences: A056239 A056240 A056241 this_sequence A056243 A056244 A056245
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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Colin L. Mallows (colinm(AT)research.avayalabs.com), Aug 23 2000
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 27 2004
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