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 A374631 #21 Jul 26 2024 19:11:49
%S A374631 1,1,2,3,6,10,19,34,63,116,218,405,763,1436,2714,5127,9718,18422,
%T A374631 34968,66397,126168,239820,456027,867325,1649970,3139288,5973746,
%U A374631 11368487,21636909,41182648,78389204,149216039,284046349,540722066,1029362133,1959609449
%N A374631 Number of integer compositions of n whose leaders of weakly increasing runs are identical.
%C A374631 The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%H A374631 Alois P. Heinz, <a href="/A374631/b374631.txt">Table of n, a(n) for n = 0..750</a> (first 101 terms from John Tyler Rascoe)
%H A374631 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374631 G.f.: 1 + Sum_{i>0} A(x,i) where A(x,i) = 1/(1-x^i) * (x^i + Sum_{z>0} ( ((x^i)/(1-x^i) * (-1 + Product_{j>i} (1/(1-x^j))))^z )) is the g.f. for compositions of this kind with all leaders equal to i. - _John Tyler Rascoe_, Jul 25 2024
%e A374631 The composition (1,3,1,4,1,2,2,1) has maximal weakly increasing subsequences ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so is counted under a(15).
%e A374631 The a(0) = 1 through a(6) = 19 compositions:
%e A374631 () (1) (2) (3) (4) (5) (6)
%e A374631 (11) (12) (13) (14) (15)
%e A374631 (111) (22) (23) (24)
%e A374631 (112) (113) (33)
%e A374631 (121) (122) (114)
%e A374631 (1111) (131) (123)
%e A374631 (1112) (141)
%e A374631 (1121) (222)
%e A374631 (1211) (1113)
%e A374631 (11111) (1122)
%e A374631 (1131)
%e A374631 (1212)
%e A374631 (1221)
%e A374631 (1311)
%e A374631 (11112)
%e A374631 (11121)
%e A374631 (11211)
%e A374631 (12111)
%e A374631 (111111)
%t A374631 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,LessEqual]&]],{n,0,15}]
%o A374631 (PARI)
%o A374631 C_x(N) = {my(x='x+O('x^N), h=1+sum(i=1,N, 1/(1-x^i)*(x^i+sum(z=1,N-i+1, (x^i/(1-x^i)*(-1+(1/prod(j=i+1,N-i,1-x^j))))^z)))); Vec(h)}
%o A374631 C_x(40) \\ _John Tyler Rascoe_, Jul 25 2024
%Y A374631 Ranked by A374633 = positions of identical rows in A374629 (sums A374630).
%Y A374631 Types of runs (instead of weakly increasing):
%Y A374631 - For leaders of identical runs we have A000005 for n > 0, ranks A272919.
%Y A374631 - For leaders of anti-runs we have A374517, ranks A374519.
%Y A374631 - For leaders of strictly increasing runs we have A374686, ranks A374685.
%Y A374631 - For leaders of weakly decreasing runs we have A374742, ranks A374744.
%Y A374631 - For leaders of strictly decreasing runs we have A374760, ranks A374759.
%Y A374631 Types of run-leaders (instead of identical):
%Y A374631 - For strictly decreasing leaders we appear to have A188920.
%Y A374631 - For weakly decreasing leaders we appear to have A189076.
%Y A374631 - For distinct leaders we have A374632, ranks A374768.
%Y A374631 - For strictly increasing leaders we have A374634.
%Y A374631 - For weakly increasing leaders we have A374635.
%Y A374631 A003242 counts anti-run compositions.
%Y A374631 A011782 counts compositions.
%Y A374631 A238130, A238279, A333755 count compositions by number of runs.
%Y A374631 A274174 counts contiguous compositions, ranks A374249.
%Y A374631 A335456 counts patterns matched by compositions.
%Y A374631 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374631 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A374631 Cf. A000009, A106356, A124766, A238343, A261982, A333213, A373949, A374518, A374687, A374743, A374761.
%K A374631 nonn
%O A374631 0,3
%A A374631 _Gus Wiseman_, Jul 23 2024
%E A374631 a(26) onwards from _John Tyler Rascoe_, Jul 25 2024