

A357136


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.


39



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, 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, 36, 0, 9, 0, 1, 0, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,8


COMMENTS

A composition of n is a finite sequence of positive integers summing to n.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (1)^(i1) y_i.


LINKS



EXAMPLE

Triangle begins:
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 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 36 0 9 0 1
0 252 0 210 0 120 0 45 0 10 0 1
462 0 462 0 330 0 165 0 55 0 11 0 1
0 924 0 792 0 495 0 220 0 66 0 12 0 1
For example, row n = 5 counts the following compositions:
. (32) . (41) . (5)
(122) (113)
(221) (212)
(1121) (311)
(2111)
(11111)


MATHEMATICA

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


CROSSREFS

The full triangle counting compositions by alternating sum is A097805.
This is the righthalf of evenindexed rows of A260492.
The triangle without top row and left column is A108044.
Ranking and counting compositions:
 length = absolute value of alternating sum: A357185, counted by A357183.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.


KEYWORD



AUTHOR



STATUS

approved



