A357136 - OEIS

%I #13 Nov 02 2023 07:54:50

%S 1,0,1,1,0,1,0,2,0,1,3,0,3,0,1,0,6,0,4,0,1,10,0,10,0,5,0,1,0,20,0,15,

%T 0,6,0,1,35,0,35,0,21,0,7,0,1,0,70,0,56,0,28,0,8,0,1,126,0,126,0,84,0,

%U 36,0,9,0,1,0,252,0,210,0,120,0,45,0,10,0,1

%N Triangle read by rows where T(n,k) is the number of integer compositions of n with alternating sum k = 0..n. Part of the full triangle A097805.

%C A composition of n is a finite sequence of positive integers summing to n.

%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

%e Triangle begins:

%e     1

%e     0   1

%e     1   0   1

%e     0   2   0   1

%e     3   0   3   0   1

%e     0   6   0   4   0   1

%e    10   0  10   0   5   0   1

%e     0  20   0  15   0   6   0   1

%e    35   0  35   0  21   0   7   0   1

%e     0  70   0  56   0  28   0   8   0   1

%e   126   0 126   0  84   0  36   0   9   0   1

%e     0 252   0 210   0 120   0  45   0  10   0   1

%e   462   0 462   0 330   0 165   0  55   0  11   0   1

%e     0 924   0 792   0 495   0 220   0  66   0  12   0   1

%e For example, row n = 5 counts the following compositions:

%e   .  (32)     .  (41)   .  (5)

%e      (122)       (113)

%e      (221)       (212)

%e      (1121)      (311)

%e      (2111)

%e      (11111)

%t Prepend[Table[If[EvenQ[nn],Prepend[#,0],#]&[Riffle[Table[Binomial[nn,k],{k,Floor[nn/2],nn}],0]],{nn,0,10}],{1}]

%Y The full triangle counting compositions by alternating sum is A097805.

%Y The version for partitions is A103919, full triangle A344651.

%Y This is the right-half of even-indexed rows of A260492.

%Y The triangle without top row and left column is A108044.

%Y Ranking and counting compositions:

%Y - product = sum: A335404, counted by A335405.

%Y - sum = twice alternating sum: A348614, counted by A262977.

%Y - length = alternating sum: A357184, counted by A357182.

%Y - length = absolute value of alternating sum: A357185, counted by A357183.

%Y A003242 counts anti-run compositions, ranked by A333489.

%Y A011782 counts compositions.

%Y A025047 counts alternating compositions, ranked by A345167.

%Y A032020 counts strict compositions, ranked by A233564.

%Y A124754 gives alternating sums of standard compositions.

%Y A238279 counts compositions by sum and number of maximal runs.

%Y Cf. A000120, A051159, A070939, A114220, A114901, A242882, A262046.

%K nonn,easy,tabl

%O 0,8

%A _Gus Wiseman_, Sep 30 2022