This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357136 #13 Nov 02 2023 07:54:50
%S A357136 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 A357136 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 A357136 36,0,9,0,1,0,252,0,210,0,120,0,45,0,10,0,1
%N 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.
%C A357136 A composition of n is a finite sequence of positive integers summing to n.
%C A357136 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e A357136 Triangle begins:
%e A357136 1
%e A357136 0 1
%e A357136 1 0 1
%e A357136 0 2 0 1
%e A357136 3 0 3 0 1
%e A357136 0 6 0 4 0 1
%e A357136 10 0 10 0 5 0 1
%e A357136 0 20 0 15 0 6 0 1
%e A357136 35 0 35 0 21 0 7 0 1
%e A357136 0 70 0 56 0 28 0 8 0 1
%e A357136 126 0 126 0 84 0 36 0 9 0 1
%e A357136 0 252 0 210 0 120 0 45 0 10 0 1
%e A357136 462 0 462 0 330 0 165 0 55 0 11 0 1
%e A357136 0 924 0 792 0 495 0 220 0 66 0 12 0 1
%e A357136 For example, row n = 5 counts the following compositions:
%e A357136 . (32) . (41) . (5)
%e A357136 (122) (113)
%e A357136 (221) (212)
%e A357136 (1121) (311)
%e A357136 (2111)
%e A357136 (11111)
%t A357136 Prepend[Table[If[EvenQ[nn],Prepend[#,0],#]&[Riffle[Table[Binomial[nn,k],{k,Floor[nn/2],nn}],0]],{nn,0,10}],{1}]
%Y A357136 The full triangle counting compositions by alternating sum is A097805.
%Y A357136 The version for partitions is A103919, full triangle A344651.
%Y A357136 This is the right-half of even-indexed rows of A260492.
%Y A357136 The triangle without top row and left column is A108044.
%Y A357136 Ranking and counting compositions:
%Y A357136 - product = sum: A335404, counted by A335405.
%Y A357136 - sum = twice alternating sum: A348614, counted by A262977.
%Y A357136 - length = alternating sum: A357184, counted by A357182.
%Y A357136 - length = absolute value of alternating sum: A357185, counted by A357183.
%Y A357136 A003242 counts anti-run compositions, ranked by A333489.
%Y A357136 A011782 counts compositions.
%Y A357136 A025047 counts alternating compositions, ranked by A345167.
%Y A357136 A032020 counts strict compositions, ranked by A233564.
%Y A357136 A124754 gives alternating sums of standard compositions.
%Y A357136 A238279 counts compositions by sum and number of maximal runs.
%Y A357136 Cf. A000120, A051159, A070939, A114220, A114901, A242882, A262046.
%K A357136 nonn,easy,tabl
%O A357136 0,8
%A A357136 _Gus Wiseman_, Sep 30 2022