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 A374760 #14 Jul 31 2024 17:27:29
%S A374760 1,1,2,3,4,6,8,11,15,21,28,38,52,70,95,129,173,234,318,428,579,784,
%T A374760 1059,1433,1942,2630,3564,4835,6559,8902,12094,16432,22340,30392,
%U A374760 41356,56304,76692,104499,142448,194264,265015,361664,493749,674278,921113,1258717
%N A374760 Number of integer compositions of n whose leaders of strictly decreasing runs are identical.
%C A374760 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%H A374760 Andrew Howroyd, <a href="/A374760/b374760.txt">Table of n, a(n) for n = 0..1000</a>
%H A374760 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374760 G.f.: 1 + Sum_{k>=1} -1 + 1/(1 - x^k*Product_{j=1..k-1} (1 + x^j)). - _Andrew Howroyd_, Jul 31 2024
%e A374760 The composition (3,3,2,1,3,2,1) has strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so is counted under a(15).
%e A374760 The a(0) = 1 through a(8) = 15 compositions:
%e A374760 () (1) (2) (3) (4) (5) (6) (7) (8)
%e A374760 (11) (21) (22) (32) (33) (43) (44)
%e A374760 (111) (31) (41) (42) (52) (53)
%e A374760 (1111) (212) (51) (61) (62)
%e A374760 (221) (222) (313) (71)
%e A374760 (11111) (321) (331) (323)
%e A374760 (2121) (421) (332)
%e A374760 (111111) (2122) (431)
%e A374760 (2212) (521)
%e A374760 (2221) (2222)
%e A374760 (1111111) (3131)
%e A374760 (21212)
%e A374760 (21221)
%e A374760 (22121)
%e A374760 (11111111)
%t A374760 Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@First/@Split[#,Greater]&]],{n,0,15}]
%o A374760 (PARI) seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=1, min(n-k,k-1), 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ _Andrew Howroyd_, Jul 31 2024
%Y A374760 For partitions instead of compositions we have A034296.
%Y A374760 The weak version is A374742, ranks A374744.
%Y A374760 The opposite version is A374686, ranks A374685.
%Y A374760 The weak opposite version is A374631, ranks A374633.
%Y A374760 Ranked by A374759.
%Y A374760 Other types of runs (instead of strictly decreasing):
%Y A374760 - For leaders of identical runs we have A000005 for n > 0, ranks A272919.
%Y A374760 - For leaders of anti-runs we have A374517, ranks A374519.
%Y A374760 Other types of run-leaders (instead of identical):
%Y A374760 - For distinct leaders we have A374761, ranks A374767.
%Y A374760 - For strictly increasing leaders we have A374762.
%Y A374760 - For strictly decreasing leaders we have A374763.
%Y A374760 - For weakly increasing leaders we have A374764.
%Y A374760 - For weakly decreasing leaders we have A374765.
%Y A374760 A003242 counts anti-run compositions, ranks A333489.
%Y A374760 A011782 counts compositions.
%Y A374760 A238130, A238279, A333755 count compositions by number of runs.
%Y A374760 A274174 counts contiguous compositions, ranks A374249.
%Y A374760 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374760 Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374640, A374761.
%K A374760 nonn
%O A374760 0,3
%A A374760 _Gus Wiseman_, Jul 29 2024
%E A374760 a(24) onwards from _Andrew Howroyd_, Jul 31 2024