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 A374517 #11 Aug 16 2024 19:05:17
%S A374517 1,1,2,4,7,13,25,46,85,160,301,561,1056,1984,3730,7037,13273,25056,
%T A374517 47382,89666,169833,322038,611128,1160660,2206219,4196730,7988731,
%U A374517 15217557,29005987,55321015,105570219,201569648,385059094,735929616,1407145439,2691681402
%N A374517 Number of integer compositions of n whose leaders of anti-runs are identical.
%C A374517 The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
%H A374517 John Tyler Rascoe, <a href="/A374517/b374517.txt">Table of n, a(n) for n = 0..100</a>
%H A374517 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374517 G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - _John Tyler Rascoe_, Aug 16 2024
%e A374517 The a(0) = 1 through a(5) = 13 compositions:
%e A374517 () (1) (2) (3) (4) (5)
%e A374517 (11) (12) (13) (14)
%e A374517 (21) (22) (23)
%e A374517 (111) (31) (32)
%e A374517 (112) (41)
%e A374517 (121) (113)
%e A374517 (1111) (131)
%e A374517 (212)
%e A374517 (221)
%e A374517 (1112)
%e A374517 (1121)
%e A374517 (1211)
%e A374517 (11111)
%t A374517 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]
%o A374517 (PARI)
%o A374517 C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
%o A374517 A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
%o A374517 B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
%o A374517 D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
%o A374517 D_x(30) \\ _John Tyler Rascoe_, Aug 16 2024
%Y A374517 For partitions instead of compositions we have A034296 or A115029.
%Y A374517 These compositions have ranks A374519.
%Y A374517 The complement is counted by A374640.
%Y A374517 Other types of runs (instead of anti-):
%Y A374517 - For leaders of identical runs we have A000005 for n > 0, ranks A272919.
%Y A374517 - For leaders of weakly increasing runs we have A374631, ranks A374633.
%Y A374517 - For leaders of strictly increasing runs we have A374686, ranks A374685.
%Y A374517 - For leaders of weakly decreasing runs we have A374742, ranks A374741.
%Y A374517 - For leaders of strictly decreasing runs we have A374760, ranks A374759.
%Y A374517 Other types of run-leaders (instead of identical):
%Y A374517 - For distinct leaders we have A374518.
%Y A374517 - For weakly increasing leaders we have A374681.
%Y A374517 - For strictly increasing leaders we have A374679.
%Y A374517 - For weakly decreasing leaders we have A374682.
%Y A374517 - For strictly decreasing leaders we have A374680.
%Y A374517 A003242 counts anti-runs, ranks A333489.
%Y A374517 A106356 counts compositions by number of maximal anti-runs.
%Y A374517 A238279 counts compositions by number of maximal runs
%Y A374517 A238424 counts partitions whose first differences are an anti-run.
%Y A374517 A274174 counts contiguous compositions, ranks A374249.
%Y A374517 Cf. A188920, A189076, A238343, A333213, A373949, A374515, A374518.
%K A374517 nonn
%O A374517 0,3
%A A374517 _Gus Wiseman_, Aug 01 2024
%E A374517 a(26) onwards from _John Tyler Rascoe_, Aug 16 2024