OFFSET
1,3
COMMENTS
Generalized Riordan array (1/(1-x), x/(1-x) + x*dif(x/1-x),x)). - Paul Barry, Dec 26 2007
Reversal of A117317. - Philippe Deléham, 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. - 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
From Gus Wiseman, Mar 06 2020: (Start)
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
LINKS
Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
Finn Bjarne Jost, Tautological Intersection Numbers and Order-Consecutive Partition Sequences, arXiv:2307.15825 [math.CO], 2023. See p. 9.
V. Strehl, Combinatoire rétrospective et créative, an on-line presentation, slide 36, SLC 71, Bertinoro,, September 18, 2013.
Volker Strehl, Lacunary Laguerre Series from a Combinatorial Perspective, Séminaire Lotharingien de Combinatoire, B76c (2017).
FORMULA
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
EXAMPLE
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;
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);
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}]](* Jean-François Alcover, Nov 17 2011 *)
PROG
(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]))
-- Reinhard Zumkeller, May 08 2014
CROSSREFS
KEYWORD
AUTHOR
Colin Mallows, Aug 23 2000
STATUS
approved