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A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

%I A351018 #16 Feb 12 2022 20:28:47
%S A351018 1,1,2,3,6,9,18,27,46,77,122,191,326,497,786,1207,1942,2905,4498,6703,
%T A351018 10574,15597,23754,35043,52422,78369,115522,169499,248150,360521,
%U A351018 532466,768275,1116126,1606669,2314426,3301879,4777078,6772657,9677138,13688079,19406214
%N A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.
%C A351018 Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).
%H A351018 Andrew Howroyd, <a href="/A351018/b351018.txt">Table of n, a(n) for n = 0..1000</a>
%H A351018 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%F A351018 a(n>0) = A351016(n)/2.
%F A351018 G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - _Andrew Howroyd_, Feb 11 2022
%e A351018 The a(1) = 1 through a(6) = 18 compositions:
%e A351018   (1)  (2)    (3)    (4)      (5)      (6)
%e A351018        (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
%e A351018               (2,1)  (2,2)    (2,3)    (2,4)
%e A351018                      (3,1)    (3,2)    (3,3)
%e A351018                      (1,1,2)  (4,1)    (4,2)
%e A351018                      (2,1,1)  (1,1,3)  (5,1)
%e A351018                               (1,2,2)  (1,1,4)
%e A351018                               (2,2,1)  (1,2,3)
%e A351018                               (3,1,1)  (1,3,2)
%e A351018                                        (2,1,3)
%e A351018                                        (2,3,1)
%e A351018                                        (3,1,2)
%e A351018                                        (3,2,1)
%e A351018                                        (4,1,1)
%e A351018                                        (1,1,2,2)
%e A351018                                        (1,2,2,1)
%e A351018                                        (2,1,1,2)
%e A351018                                        (2,2,1,1)
%t A351018 Table[Length[Select[Tuples[{0,1},n],#=={}||First[#]==1&&UnsameQ@@Split[#]&]],{n,0,10}]
%o A351018 (PARI) P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
%o A351018 seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p,k\2,y)*(k\2)!*polcoef(p,(k+1)\2,y)*((k+1)\2)!)) \\ _Andrew Howroyd_, Feb 11 2022
%Y A351018 The case of partitions is A000726.
%Y A351018 The version for run-lengths instead of runs is A032020.
%Y A351018 These words are ranked by A175413.
%Y A351018 A005811 counts runs in binary expansion.
%Y A351018 A011782 counts integer compositions.
%Y A351018 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351018 A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
%Y A351018 A116608 counts compositions by number of distinct parts.
%Y A351018 A238130 and A238279 count compositions by number of runs.
%Y A351018 A242882 counts compositions with distinct multiplicities.
%Y A351018 A297770 counts distinct runs in binary expansion.
%Y A351018 A325545 counts compositions with distinct differences.
%Y A351018 A329738 counts compositions with equal run-lengths.
%Y A351018 A329744 counts compositions by runs-resistance.
%Y A351018 A351014 counts distinct runs in standard compositions.
%Y A351018 Counting words with all distinct runs:
%Y A351018 - A351013 = compositions, for run-lengths A329739, ranked by A351290.
%Y A351018 - A351016 = binary words, for run-lengths A351017.
%Y A351018 - A351200 = patterns, for run-lengths A351292.
%Y A351018 - A351202 = permutations of prime factors.
%Y A351018 Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.
%K A351018 nonn
%O A351018 0,3
%A A351018 _Gus Wiseman_, Feb 09 2022
%E A351018 Terms a(21) and beyond from _Andrew Howroyd_, Feb 11 2022