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 A129838 Number of up/down (or down/up) compositions of n into distinct parts. 6
 1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 18, 21, 30, 38, 52, 78, 97, 128, 170, 222, 285, 421, 510, 683, 872, 1148, 1440, 1893, 2576, 3209, 4151, 5313, 6784, 8615, 10969, 13755, 18573, 22713, 29173, 36536, 46705, 57899, 73696, 91076, 114777, 148531, 182813, 228938, 287042 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Original name was: Number of alternating compositions of n into distinct parts. A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. - Gus Wiseman, Jan 15 2022 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 FORMULA G.f.: Sum_{k>=0} A000111(k)*x^(k*(k+1)/2)/Product_{i=1..k} (1-x^i). - Vladeta Jovovic, May 24 2007 a(n) = Sum_{k=0..A003056(n)} A000111(k) * A008289(n,k). - Alois P. Heinz, Dec 22 2021 a(n) = (A349054(n) + 1)/2. - Gus Wiseman, Jan 15 2022 EXAMPLE From Gus Wiseman, Jan 15 2022: (Start) The a(1) = 1 through a(8) = 8 up/down strict compositions (non-strict A025048): (1) (2) (3) (4) (5) (6) (7) (8) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (2,3) (2,4) (2,5) (2,6) (1,3,2) (3,4) (3,5) (2,3,1) (1,4,2) (1,4,3) (2,4,1) (1,5,2) (2,5,1) (3,4,1) The a(1) = 1 through a(8) = 8 down/up strict compositions (non-strict A025049): (1) (2) (3) (4) (5) (6) (7) (8) (2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (4,1) (5,1) (5,2) (6,2) (2,1,3) (6,1) (7,1) (3,1,2) (2,1,4) (2,1,5) (4,1,2) (3,1,4) (4,1,3) (5,1,2) (End) MAPLE g:= proc(u, o) option remember; `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u)) end: b:= proc(n, k) option remember; `if`(k<0 or n<0, 0, `if`(k=0, `if`(n=0, 1, 0), b(n-k, k)+b(n-k, k-1))) end: a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2)): seq(a(n), n=0..60); # Alois P. Heinz, Dec 22 2021 MATHEMATICA whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]y[[m+1]]], {m, 1, Length[y]-1}]; Table[Length[Select[Join@@Permutations/@ Select[IntegerPartitions[n], UnsameQ@@#&], whkQ]], {n, 0, 15}] (* Gus Wiseman, Jan 15 2022 *) CROSSREFS The case of permutations is A000111. This is the up/down case of A032020. This is the strict case of A129852/A129853, strong A025048/A025049. The undirected version is A349054. A001250 = alternating permutations, complement A348615. A003242 = Carlitz compositions, complement A261983. A011782 = compositions, unordered A000041. A025047 = alternating compositions, complement A345192. A349052 = weakly alternating compositions, complement A349053. Cf. A003056, A008289, A008965, A015723, A072706, A128761, A218074, A345165, A345170, A345195, A349800. Sequence in context: A076571 A084783 A265853 * A032153 A309223 A116465 Adjacent sequences: A129835 A129836 A129837 * A129839 A129840 A129841 KEYWORD easy,nonn AUTHOR Vladeta Jovovic, May 21 2007 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Dec 22 2021 Name changed from "alternating" to "up/down" by Gus Wiseman, Jan 15 2022 STATUS approved

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Last modified April 1 09:50 EDT 2023. Contains 361688 sequences. (Running on oeis4.)