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%I #26 Feb 21 2023 13:23:17
%S 1,1,2,3,6,8,11,16,22,28,39,49,61,77,93,114,140,169,198,233,276,321,
%T 381,439,509,591,678,774,883,1007,1147,1300,1465,1641,1845,2068,2317,
%U 2590,2881,3193,3549,3928,4341,4793,5282,5813,6401,7027,7699,8432,9221,10076
%N Number of compositions of n with strictly increasing first quotients.
%C 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).
%H Alois P. Heinz, <a href="/A342493/b342493.txt">Table of n, a(n) for n = 0..500</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e The composition (3,1,1,2) has first quotients (1/3,1,2) so is counted under a(7).
%e The a(1) = 1 through a(7) = 16 compositions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
%e (2,1) (2,2) (2,3) (2,4) (2,5)
%e (3,1) (3,2) (3,3) (3,4)
%e (1,1,2) (4,1) (4,2) (4,3)
%e (2,1,1) (1,1,3) (5,1) (5,2)
%e (2,1,2) (1,1,4) (6,1)
%e (3,1,1) (2,1,3) (1,1,5)
%e (3,1,2) (2,1,4)
%e (4,1,1) (2,2,3)
%e (2,1,1,2) (3,1,3)
%e (3,2,2)
%e (4,1,2)
%e (5,1,1)
%e (2,1,1,3)
%e (3,1,1,2)
%p b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
%p `if`(q=0 or q>l/j, b(n-j, l/j, j), 0), j=1..n))
%p end:
%p a:= n-> b(n, 0$2):
%p seq(a(n), n=0..55); # _Alois P. Heinz_, Mar 25 2021
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
%t (* Second program: *)
%t b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
%t If[q == 0 || q > l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
%t a[n_] := b[n, 0, 0];
%t a /@ Range[0, 55] (* _Jean-François Alcover_, May 19 2021, after _Alois P. Heinz_ *)
%Y The version for differences instead of quotients is A325547.
%Y The weakly increasing version is A342492.
%Y The strictly decreasing version is A342494.
%Y The unordered version is A342498, ranked by A342524.
%Y The strict unordered version is A342517.
%Y A000005 counts constant compositions.
%Y A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A001055 counts factorizations.
%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A074206 counts ordered factorizations.
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A274199 counts compositions with all adjacent parts x < 2y.
%Y Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 16 2021
%E a(21)-a(51) from _Alois P. Heinz_, Mar 18 2021
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