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 A374746 #15 Dec 30 2024 21:53:04
%S A374746 1,1,2,3,5,7,12,18,31,51,86,143,241,397,657,1082,1771,2889,4697,7605,
%T A374746 12269,19720,31580,50412,80205,127208,201149,317171,498717,782076,
%U A374746 1223230,1908381,2969950,4610949,7141972,11037276,17019617,26188490,40213388,61624824
%N A374746 Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.
%C A374746 The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
%H A374746 Andrew Howroyd, <a href="/A374746/b374746.txt">Table of n, a(n) for n = 0..1000</a>
%H A374746 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374746 G.f.: Sum_{k>=0} x^k*Q(k,x)/(1 - x^k) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 - x^k/(1 - x^k) + x^k*Product_{j=1..k} (1 - x^j))/(1 - x^k) for k > 0. - _Andrew Howroyd_, Dec 30 2024
%e A374746 The a(0) = 1 through a(7) = 18 compositions:
%e A374746 () (1) (2) (3) (4) (5) (6) (7)
%e A374746 (11) (21) (22) (32) (33) (43)
%e A374746 (111) (31) (41) (42) (52)
%e A374746 (211) (221) (51) (61)
%e A374746 (1111) (311) (222) (322)
%e A374746 (2111) (312) (331)
%e A374746 (11111) (321) (412)
%e A374746 (411) (421)
%e A374746 (2211) (511)
%e A374746 (3111) (2221)
%e A374746 (21111) (3112)
%e A374746 (111111) (3121)
%e A374746 (3211)
%e A374746 (4111)
%e A374746 (22111)
%e A374746 (31111)
%e A374746 (211111)
%e A374746 (1111111)
%t A374746 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
%o A374746 (PARI) seq(n)={my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q/(1-x^k); p /= 1 - x^k; q *= (1 - x^k/(1-x^k) + x^k*p)/(1-x^k) ); Vec(r + x^(n\2+1)*q/(1-x))} \\ _Andrew Howroyd_, Dec 30 2024
%Y A374746 Ranked by positions of strictly decreasing rows in A374740, opp. A374629.
%Y A374746 Types of runs (instead of weakly decreasing):
%Y A374746 - For leaders of identical runs we have A000041.
%Y A374746 - For leaders of weakly increasing runs we have A188920.
%Y A374746 - For leaders of anti-runs we have A374680.
%Y A374746 - For leaders of strictly increasing runs we have A374689.
%Y A374746 - For leaders of strictly decreasing runs we have A374763.
%Y A374746 Types of run-leaders (instead of strictly decreasing):
%Y A374746 - For weakly increasing leaders we appear to have A188900.
%Y A374746 - For identical leaders we have A374742.
%Y A374746 - For distinct leaders we have A374743, ranks A374701.
%Y A374746 - For strictly increasing leaders we have opposite A374634.
%Y A374746 - For weakly decreasing leaders we have A374747.
%Y A374746 A011782 counts compositions.
%Y A374746 A238130, A238279, A333755 count compositions by number of runs.
%Y A374746 A335456 counts patterns matched by compositions.
%Y A374746 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374746 A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y A374746 Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.
%K A374746 nonn
%O A374746 0,3
%A A374746 _Gus Wiseman_, Jul 26 2024
%E A374746 a(24)-a(39) from _Alois P. Heinz_, Jul 26 2024