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 A228524 #29 Sep 22 2013 16:23:00 %S A228524 1,1,1,3,1,1,7,3,1,1,16,7,3,1,1,36,16,7,3,1,1,80,36,16,7,3,1,1,176,80, %T A228524 36,16,7,3,1,1,384,176,80,36,16,7,3,1,1,832,384,176,80,36,16,7,3,1,1, %U A228524 1792,832,384,176,80,36,16,7,3,1,1,3840,1792,832,384,176,80,36,16,7,3,1,1 %N A228524 Triangle read by rows: T(n,k) = total number of occurrences of parts k in the n-th section of the set of compositions (ordered partitions) of any integer >= n. %C A228524 Here, define "n-th section of the set of compositions of any integer >= n" to be the set formed by all parts that occur as a result of taking all compositions (ordered partitions) of n and then remove all parts of the compositions of n-1, if n >= 1. Hence the n-th section of the set of compositions of any integer >= n is also the last section of the set of compositions of n. Note that by definition the ordering of compositions is not relevant. For the visualization of the sections here we use a dissection of the diagram of compositions of n in colexicographic order, see example. %C A228524 The equivalent sequence for partitions is A182703. %C A228524 Row n lists the first n terms of A045891 in decreasing order. %F A228524 T(n,k) = A045891(n-k), n >= 1, 1<=k<=n. %e A228524 Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4, also the first four sections of the set of compositions of any integer >= 4: %e A228524 . %e A228524 . 1 2 3 4 %e A228524 . _ _ _ _ %e A228524 . |_| _| | | | | | %e A228524 . |_ _| _ _| | | | %e A228524 . |_| | | | %e A228524 . |_ _ _| _ _ _| | %e A228524 . |_| | | %e A228524 . |_ _| | %e A228524 . |_| | %e A228524 . |_ _ _ _| %e A228524 . %e A228524 For n = 4 and k = 2, T(4,2) = 3 because there are 3 parts of size 2 in all compositions of 4, see below: %e A228524 -------------------------------------------------------- %e A228524 . The last section Number of %e A228524 . Composition of 4 of the set of parts of %e A228524 . compositions of 4 size k %e A228524 . -------------------- ------------------- %e A228524 . Diagram Diagram k = 1 2 3 4 %e A228524 . ------------------------------------------------------ %e A228524 . _ _ _ _ _ %e A228524 . 1+1+1+1 |_| | | | 1 | | 1 0 0 0 %e A228524 . 2+1+1 |_ _| | | 1 | | 1 0 0 0 %e A228524 . 1+2+1 |_| | | 1 | | 1 0 0 0 %e A228524 . 3+1 |_ _ _| | 1 _ _ _| | 1 0 0 0 %e A228524 . 1+1+2 |_| | | 1+1+2 |_| | | 2 1 0 0 %e A228524 . 2+2 |_ _| | 2+2 |_ _| | 0 2 0 0 %e A228524 . 1+3 |_| | 1+3 |_| | 1 0 1 0 %e A228524 . 4 |_ _ _ _| 4 |_ _ _ _| 0 0 0 1 %e A228524 . --------- %e A228524 . Column sums give row 4: 7,3,1,1 %e A228524 . %e A228524 Triangle begins: %e A228524 1; %e A228524 1, 1; %e A228524 3, 1, 1; %e A228524 7, 3, 1, 1; %e A228524 16, 7, 3, 1, 1; %e A228524 36, 16, 7, 3, 1, 1; %e A228524 80, 36, 16, 7, 3, 1, 1; %e A228524 176, 80, 36, 16, 7, 3, 1, 1; %e A228524 384, 176, 80, 36, 16, 7, 3, 1, 1; %e A228524 832, 384, 176, 80, 36, 16, 7, 3, 1, 1; %e A228524 1792, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1; %e A228524 3840, 1792, 832, 384,176, 80, 36, 16, 7, 3, 1, 1; %e A228524 8192, 3840,1792, 832,384,176, 80, 36, 16, 7, 3, 1, 1; %e A228524 ... %Y A228524 Row sums give A045623. Every column gives A045891. %Y A228524 Cf. A011782, A182703, A221876, A228350, A228366, A228367, A228370, A228526, A228527. %K A228524 nonn,tabl %O A228524 1,4 %A A228524 _Omar E. Pol_, Aug 26 2013