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 A332670 #15 Feb 16 2025 08:33:59
%S A332670 1,0,1,0,1,1,0,1,2,1,0,1,3,2,1,0,1,4,5,2,1,0,1,5,7,5,2,1,0,1,6,11,10,
%T A332670 5,2,1,0,1,7,15,16,10,5,2,1,0,1,8,20,24,20,10,5,2,1,0,1,9,25,36,31,20,
%U A332670 10,5,2,1,0,1,10,32,50,50,36,20,10,5,2,1
%N A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.
%C A332670 A composition of n is a finite sequence of positive integers summing to n.
%C A332670 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332670 Andrew Howroyd, <a href="/A332670/b332670.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A332670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F A332670 G.f.: A(x,y) = 1 + Sum_{j>0} y*x^j/((1 - y*x^j)*Product_{k>j} (1 - y*x^k)^2). - _Andrew Howroyd_, Jan 11 2024
%e A332670 Triangle begins:
%e A332670 1
%e A332670 0 1
%e A332670 0 1 1
%e A332670 0 1 2 1
%e A332670 0 1 3 2 1
%e A332670 0 1 4 5 2 1
%e A332670 0 1 5 7 5 2 1
%e A332670 0 1 6 11 10 5 2 1
%e A332670 0 1 7 15 16 10 5 2 1
%e A332670 0 1 8 20 24 20 10 5 2 1
%e A332670 0 1 9 25 36 31 20 10 5 2 1
%e A332670 0 1 10 32 50 50 36 20 10 5 2 1
%e A332670 0 1 11 38 67 73 59 36 20 10 5 2 1
%e A332670 Column n = 7 counts the following compositions:
%e A332670 (7) (16) (115) (1114) (11113) (111112) (1111111)
%e A332670 (25) (124) (1123) (11122) (211111)
%e A332670 (34) (133) (1222) (21112)
%e A332670 (43) (214) (2113) (22111)
%e A332670 (52) (223) (2122) (31111)
%e A332670 (61) (313) (2212)
%e A332670 (322) (2221)
%e A332670 (331) (3112)
%e A332670 (412) (3211)
%e A332670 (421) (4111)
%e A332670 (511)
%t A332670 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332670 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],unimodQ[-#]&]],{n,0,10},{k,0,n}]
%o A332670 (PARI) T(n)={[Vecrev(p) | p<-Vec(1 + sum(j=1, n, y*x^j/((1-y*x^j) * prod(k=j+1, n-j, 1 - y*x^k + O(x*x^(n-j)))^2)))]}
%o A332670 { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 11 2024
%Y A332670 The case of partitions is A072233.
%Y A332670 Dominated by A072704 (the non-negated version).
%Y A332670 The strict case is A072705.
%Y A332670 The case of constant compositions is A113704.
%Y A332670 Row sums are A332578.
%Y A332670 Unimodal compositions are A001523.
%Y A332670 Unimodal normal sequences appear to be A007052.
%Y A332670 Non-unimodal compositions are A115981.
%Y A332670 Non-unimodal normal sequences are A328509.
%Y A332670 Numbers whose negated unsorted prime signature is not unimodal are A332282.
%Y A332670 Partitions whose negated run-lengths are unimodal are A332638.
%Y A332670 Compositions whose negation is not unimodal are A332669.
%Y A332670 Partitions whose negated 0-appended first differences are unimodal: A332728.
%Y A332670 Cf. A011782, A107429, A227038, A332280, A332283, A332639, A332642, A332741, A332742, A332744, A332832, A332870.
%K A332670 nonn,tabl
%O A332670 0,9
%A A332670 _Gus Wiseman_, Feb 29 2020