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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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13
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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
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OFFSET
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1,3
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COMMENTS
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Generalized Riordan array (1/(1-x), x/(1-x) + x*dif(x/1-x),x)). - Paul Barry, Dec 26 2007
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. - Philippe Deléham, Feb 11 2012
This sequence is given in the Strehl presentation with the o.g.f. (1-z)/[1-2(1+t)z+(1+t)z^2], with offset 0, along with a recursion relation, a combinatorial interpretation, and relations to Hermite and Laguerre polynomials. Note that the o.g.f. is related to that of A049310. - Tom Copeland, Jan 08 2017
T(n,k) is also the number of unimodal length-n sequences covering an initial interval of positive integers with maximum part k, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the sequences counted by row n = 4 are:
(1111) (1112) (1123) (1234)
(1121) (1132) (1243)
(1122) (1223) (1342)
(1211) (1231) (1432)
(1221) (1232) (2341)
(1222) (1233) (2431)
(2111) (1321) (3421)
(2211) (1322) (4321)
(2221) (1332)
(2231)
(2311)
(2321)
(2331)
(3211)
(3221)
(3321)
(End)
T(n,k) is the number of hexagonal directed-column convex polyominoes of area n with k columns (see Baril et al. at page 9). - Stefano Spezia, Oct 14 2023
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LINKS
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FORMULA
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The Hwang and Mallows reference gives explicit formulas.
T(n,k) = Sum_{j=0..k-1} (-1)^(k-1-j)*binomial(k-1, j)*binomial(n+2j-1, 2j) (1<=k<=n); this is formula (11) in the Huang and Mallows 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(2,1) = 1, T(2,2) = 2. - Philippe Deléham, Feb 11 2012
G.f.: -(-1+x)*x*y/(1-2*x-2*x*y+x^2*y+x^2). - R. J. Mathar, Aug 11 2015
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EXAMPLE
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Triangle begins:
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;
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
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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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MATHEMATICA
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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}]](* Jean-François Alcover, Nov 17 2011 *)
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PROG
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(Haskell)
a056242 n k = a056242_tabl !! (n-1)!! (k-1)
a056242_row n = a056242_tabl !! (n-1)
a056242_tabl = [1] : [1, 2] : f [1] [1, 2] where
f us vs = ws : f vs ws where
ws = zipWith (-) (map (* 2) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))
(zipWith (+) ([0] ++ us ++ [0]) (us ++ [0, 0]))
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CROSSREFS
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Ordered set-partitions are A000670.
Cf. A001523, A049310, A072704, A084938, A097805, A117317, A227038, A328509, A332294, A332673, A332724, A332872.
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KEYWORD
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AUTHOR
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STATUS
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approved
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