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A336108 Number of compositions of 2*n with n maximal runs.

%I A336108 #9 Feb 02 2021 18:21:28
%S A336108 1,2,4,14,36,99,274,813,2278,6692,19206,56687,164416,486052,1422654,
%T A336108 4214023,12408476,36825663,108926976,323856358,961177042,2862551860,
%U A336108 8518115200,25407468667,75763113682,226297498429,675951314988,2021528322571,6046881759308,18104307275968,54219605813884
%N A336108 Number of compositions of 2*n with n maximal runs.
%H A336108 Andrew Howroyd, <a href="/A336108/b336108.txt">Table of n, a(n) for n = 0..200</a>
%F A336108 a(n) = A333755(2*n,n).
%F A336108 a(n) = [x^(2*n)*y^n] (1 - y)/(1 - y - y*Sum_{d>=1} (1-y)^d*x^d/(1 - x^d)). - _Andrew Howroyd_, Feb 02 2021
%e A336108 The a(0) = 1 through a(3) = 14 compositions:
%e A336108   ()  (2)    (1,3)    (1,2,3)
%e A336108       (1,1)  (3,1)    (1,3,2)
%e A336108              (1,1,2)  (1,4,1)
%e A336108              (2,1,1)  (2,1,3)
%e A336108                       (2,3,1)
%e A336108                       (3,1,2)
%e A336108                       (3,2,1)
%e A336108                       (1,1,3,1)
%e A336108                       (1,2,2,1)
%e A336108                       (1,3,1,1)
%e A336108                       (2,1,1,2)
%e A336108                       (1,1,1,2,1)
%e A336108                       (1,1,2,1,1)
%e A336108                       (1,2,1,1,1)
%t A336108 Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#]]==n&]],{n,0,10}]
%o A336108 (PARI) a(n)={polcoef(polcoef((1 - y)/(1 - y - y*sum(d=1, 2*n, (1-y)^d*x^d/(1 - x^d) + O(x^(2*n+1)))),  2*n, x), n, y)} \\ _Andrew Howroyd_, Feb 02 2021
%Y A336108 A333755 has this as main diagonal n = 2*k.
%Y A336108 A337504 is the version for anti-runs.
%Y A336108 A337505 is the version for anti-run patterns.
%Y A336108 A337564 is the version for patterns.
%Y A336108 A003242 counts anti-run compositions.
%Y A336108 A011782 counts compositions.
%Y A336108 A106356 counts compositions by number of adjacent equal parts.
%Y A336108 A124767 counts maximal runs in standard compositions.
%Y A336108 A238343 counts compositions by descents.
%Y A336108 A272919 ranks runs.
%Y A336108 A333213 counts compositions by weak ascents.
%Y A336108 A333769 gives run-lengths of standard compositions.
%Y A336108 Cf. A066099, A124762, A235998, A238130, A238279, A333381, A333382, A333489.
%K A336108 nonn
%O A336108 0,2
%A A336108 _Gus Wiseman_, Sep 04 2020
%E A336108 Terms a(11) and beyond from _Andrew Howroyd_, Feb 02 2021