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Search: id:A056241
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| A056241 |
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Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n). |
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+0
7
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| 1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 19, 10, 1, 1, 15, 45, 45, 15, 1, 1, 21, 90, 141, 90, 21, 1, 1, 28, 161, 357, 357, 161, 28, 1, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 26 2005
Subtriangle (for 1<=k<=n)of triangle defined by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2006
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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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T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).
G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 26 2005
Sum_{k, 1<=k<=n}T(n,k)=A124302(n) . Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2006
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EXAMPLE
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1; 1,1; 1,3,1; 1,6,6,1; 1,10,19,10,1; ...
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PROGRAM
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(PARI) T(n, k)=if(n<k|k<1, 0, polcoeff((1+x+x^2)^(n-1)+O(x^(2*k)), 2*k-2)) (Hanna)
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CROSSREFS
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Columns are A000217, A005712, A005714, A005716.
Cf. A027907, A104027.
Sequence in context: A109647 A054120 A114176 this_sequence A001263 A107105 A088925
Adjacent sequences: A056238 A056239 A056240 this_sequence A056242 A056243 A056244
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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 James A. Sellers (sellersj(AT)math.psu.edu), Aug 25 2000
More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 26 2005
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