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 A374702 #12 Aug 15 2024 02:04:43
%S A374702 0,0,0,2,3,6,9,13,17,23,28,35,42,50,58,68,77,88,99,111,123,137,150,
%T A374702 165,180,196,212,230,247,266,285,305,325,347,368,391,414,438,462,488,
%U A374702 513,540,567,595,623,653,682,713,744,776,808,842,875,910,945,981
%N A374702 Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.
%C A374702 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 A374702 Andrew Howroyd, <a href="/A374702/b374702.txt">Table of n, a(n) for n = 0..10000</a>
%H A374702 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%H A374702 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A374702 G.f.: x^3*(2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3). - _Andrew Howroyd_, Aug 14 2024
%e A374702 The a(0) = 0 through a(8) = 17 compositions:
%e A374702 . . . (3) (31) (32) (33) (322) (332)
%e A374702 (12) (112) (122) (321) (331) (3221)
%e A374702 (121) (311) (1122) (1222) (3311)
%e A374702 (1112) (1221) (3211) (11222)
%e A374702 (1121) (3111) (11122) (12221)
%e A374702 (1211) (11112) (11221) (32111)
%e A374702 (11121) (12211) (111122)
%e A374702 (11211) (31111) (111221)
%e A374702 (12111) (111112) (112211)
%e A374702 (111121) (122111)
%e A374702 (111211) (311111)
%e A374702 (112111) (1111112)
%e A374702 (121111) (1111121)
%e A374702 (1111211)
%e A374702 (1112111)
%e A374702 (1121111)
%e A374702 (1211111)
%t A374702 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,GreaterEqual]]==3&]],{n,0,15}]
%o A374702 (PARI) seq(n)={Vec((2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3) + O(x^(n-2)), -n-1)} \\ _Andrew Howroyd_, Aug 14 2024
%Y A374702 The version for k = 2 is A004526.
%Y A374702 The version for partitions is A069905 or A001399 (shifted).
%Y A374702 For reversed partitions we appear to have A137719.
%Y A374702 For length instead of sum we have A241627.
%Y A374702 For leaders of constant runs we have A373952.
%Y A374702 The opposite rank statistic is A374630, row-sums of A374629.
%Y A374702 The corresponding rank statistic is A374741 row-sums of A374740.
%Y A374702 Column k = 3 of A374748.
%Y A374702 A003242 counts anti-run compositions.
%Y A374702 A011782 counts integer compositions.
%Y A374702 A238130, A238279, A333755 count compositions by number of runs.
%Y A374702 A274174 counts contiguous compositions, ranks A374249.
%Y A374702 Cf. A000009, A000041, A101271, A106356, A188900, A238343, A261982, A333213.
%K A374702 nonn,easy
%O A374702 0,4
%A A374702 _Gus Wiseman_, Aug 12 2024
%E A374702 a(27) onwards from _Andrew Howroyd_, Aug 14 2024