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 A375405 #14 Aug 21 2024 19:45:01
%S A375405 0,0,0,0,0,1,1,3,5,8,13,20,29,42,62,83,117,158,214,283,377,488,641,
%T A375405 823,1058,1345,1714,2154,2713,3387,4222,5230,6474,7959,9782,11956,
%U A375405 14591,17737,21529,26026,31422,37811,45425,54418,65097,77652,92510,109943,130468
%N A375405 Number of integer partitions of n with a repeated part other than the least.
%C A375405 Also partitions whose minima of maximal anti-runs are not identical. An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
%H A375405 Alois P. Heinz, <a href="/A375405/b375405.txt">Table of n, a(n) for n = 0..5000</a> (first 101 terms from John Tyler Rascoe)
%H A375405 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375405 G.f.: Sum_{i>0} (Sum_{j>i} ( (x^(i+(2*j)))/(1-x^i) * Product_{k>=i} (1-[k<j]*x^(2*k))/(1-x^k) )). - _John Tyler Rascoe_, Aug 21 2024
%e A375405 The a(0) = 0 through a(10) = 13 partitions:
%e A375405 . . . . . (221) (2211) (331) (332) (441) (442)
%e A375405 (2221) (3221) (3321) (3322)
%e A375405 (22111) (3311) (4221) (3331)
%e A375405 (22211) (22221) (4411)
%e A375405 (221111) (32211) (5221)
%e A375405 (33111) (32221)
%e A375405 (222111) (33211)
%e A375405 (2211111) (42211)
%e A375405 (222211)
%e A375405 (322111)
%e A375405 (331111)
%e A375405 (2221111)
%e A375405 (22111111)
%t A375405 Table[Length[Select[IntegerPartitions[n], !SameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]
%t A375405 - or -
%t A375405 Table[Length[Select[IntegerPartitions[n], !UnsameQ@@DeleteCases[#,Min@@#]&]],{n,0,30}]
%o A375405 (PARI)
%o A375405 A_x(N) = {my(x='x+O('x^N), f=sum(i=1,N,sum(j=i+1,N-i, ((x^(i+(2*j)))/(1-x^i))*prod(k=i+1,N-i-(2*j), if(k<j,1-x^(2*k),1)/(1-x^k))))); concat([0,0,0,0,0],Vec(f))}
%o A375405 A_x(60) \\ _John Tyler Rascoe_, Aug 21 2024
%Y A375405 The complement for maxima instead of minima is A034296.
%Y A375405 The complement is counted by A115029, ranks A375396.
%Y A375405 For maxima instead of minima we have A239955, ranks A073492.
%Y A375405 These partitions have ranks A375397.
%Y A375405 For distinct instead of identical we have A375404, ranks A375399.
%Y A375405 A000041 counts integer partitions, strict A000009.
%Y A375405 A003242 counts anti-run compositions, ranks A333489.
%Y A375405 A055887 counts sequences of partitions with total sum n.
%Y A375405 A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
%Y A375405 Cf. A046660, A065200, A065201, A141199, A358836, A374704, A375133, A375134, A375136, A375401.
%K A375405 nonn
%O A375405 0,8
%A A375405 _Gus Wiseman_, Aug 17 2024