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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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4
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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; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Generalized Riordan array (1/(1-x), x/(1-x)+x*dif(x/1-x),x)). - Paul Barry (pbarry(AT)wit.ie), Dec 26 2007
Reversal of A117317. - DELEHAM Philippe, Feb 11 2012
Essentially given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Feb 11 2012
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REFERENCES
| 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
| 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.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(1,1) = 1, T(1,2) = 1, T(2,2) = 2 - DELEHAM Philippe, Feb 11 2012
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EXAMPLE
| 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}.
Triangle (1, 0, 1/2, 1/2, 0, 0, 0,...) DELTA (0, 2, 0, 0, 0,...) begins:
1
1, 0
1, 2, 0
1, 5, 4, 0
1, 9, 16, 8, 0
1, 14, 41, 44, 16, 0
1, 20, 85, 146, 112, 32, 0
1, 27, 155, 377, 456, 272, 64, 0
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MAPLE
| 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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MATHEMATICA
| rows = 11; t[n_, k_] := (-1)^(k+1)*HypergeometricPFQ[{1-k, (n+1)/2, n/2}, {1/2, 1}, 1]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* From Jean-François Alcover, Nov 17 2011 *)
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CROSSREFS
| Second diagonal gives A053220.
Sequence in context: A171090 A141506 A194682 * A128718 A112358 A126351
Adjacent sequences: A056239 A056240 A056241 * A056243 A056244 A056245
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KEYWORD
| nonn,tabl,easy,nice,changed
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AUTHOR
| Colin L. Mallows (colinm(AT)research.avayalabs.com), Aug 23 2000
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 27 2004
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