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 A069916 #41 Feb 16 2025 08:32:46
%S A069916 1,1,2,4,6,9,14,20,26,36,47,60,80,102,127,159,194,236,291,355,425,514,
%T A069916 611,718,856,1009,1182,1381,1605,1861,2156,2496,2873,3299,3778,4301,
%U A069916 4902,5574,6325,7176,8116,9152,10317,11610,13028,14611,16354,18259,20365
%N A069916 Number of log-concave compositions (ordered partitions) of n.
%C A069916 These are compositions with weakly decreasing first quotients, where the first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3). - _Gus Wiseman_, Mar 16 2021
%H A069916 Alois P. Heinz, <a href="/A069916/b069916.txt">Table of n, a(n) for n = 0..300</a>
%H A069916 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a069/A069916.java">Java program</a> (github)
%H A069916 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A069916 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A069916 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A069916 Out of the 8 compositions of 4, only 2+1+1 and 1+1+2 are not log-concave, so a(4)=6.
%e A069916 From _Gus Wiseman_, Mar 15 2021: (Start)
%e A069916 The a(1) = 1 through a(6) = 14 compositions:
%e A069916 (1) (2) (3) (4) (5) (6)
%e A069916 (11) (12) (13) (14) (15)
%e A069916 (21) (22) (23) (24)
%e A069916 (111) (31) (32) (33)
%e A069916 (121) (41) (42)
%e A069916 (1111) (122) (51)
%e A069916 (131) (123)
%e A069916 (221) (132)
%e A069916 (11111) (141)
%e A069916 (222)
%e A069916 (231)
%e A069916 (321)
%e A069916 (1221)
%e A069916 (111111)
%e A069916 (End)
%t A069916 (* This program is not suitable for computing a large number of terms *)
%t A069916 compos[n_] := Permutations /@ IntegerPartitions[n] // Flatten[#, 1]&;
%t A069916 logConcaveQ[p_] := And @@ Table[p[[i]]^2 >= p[[i-1]]*p[[i+1]], {i, 2, Length[p]-1}]; a[n_] := Count[compos[n], p_?logConcaveQ]; Table[an = a[n]; Print["a(", n, ") = ", an]; a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 29 2016 *)
%t A069916 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}] (* _Gus Wiseman_, Mar 15 2021 *)
%o A069916 (Sage) def A069916(n) : return sum(all(p[i]^2 >= p[i-1] * p[i+1] for i in range(1, len(p)-1)) for p in Compositions(n)) # _Eric M. Schmidt_, Sep 29 2013
%Y A069916 The version for differences instead of quotients is A070211.
%Y A069916 A000005 counts constant compositions.
%Y A069916 A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A069916 A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A069916 A001055 counts factorizations.
%Y A069916 A002843 counts compositions with adjacent parts x <= 2y.
%Y A069916 A003238 counts chains of divisors summing to n-1, with strict case A122651.
%Y A069916 A003242 counts anti-run compositions.
%Y A069916 A074206 counts ordered factorizations.
%Y A069916 A167865 counts strict chains of divisors summing to n.
%Y A069916 Cf. A008965, A048004, A059966, A167606, A175342, A325547.
%K A069916 nonn,nice
%O A069916 0,3
%A A069916 _Pontus von Brömssen_, Apr 24 2002