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A342532 Number of even-length compositions of n with alternating parts distinct.

%I A342532 #11 Apr 17 2021 03:43:36
%S A342532 1,0,1,2,3,4,9,14,28,44,83,136,250,424,757,1310,2313,4018,7081,12314,
%T A342532 21650,37786,66264,115802,202950,354858,621525,1087252,1903668,
%U A342532 3330882,5831192,10204250,17862232,31260222,54716913,95762576,167614445,293356422,513456686
%N A342532 Number of even-length compositions of n with alternating parts distinct.
%C A342532 These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.
%H A342532 Andrew Howroyd, <a href="/A342532/b342532.txt">Table of n, a(n) for n = 0..500</a>
%F A342532 G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - _Andrew Howroyd_, Apr 16 2021
%e A342532 The a(2) = 1 through a(7) = 14 compositions:
%e A342532   (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
%e A342532          (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
%e A342532                 (3,1)  (3,2)  (3,3)      (3,4)
%e A342532                        (4,1)  (4,2)      (4,3)
%e A342532                               (5,1)      (5,2)
%e A342532                               (1,1,2,2)  (6,1)
%e A342532                               (1,2,2,1)  (1,1,2,3)
%e A342532                               (2,1,1,2)  (1,1,3,2)
%e A342532                               (2,2,1,1)  (1,2,3,1)
%e A342532                                          (1,3,2,1)
%e A342532                                          (2,1,1,3)
%e A342532                                          (2,3,1,1)
%e A342532                                          (3,1,1,2)
%e A342532                                          (3,2,1,1)
%t A342532 qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
%t A342532 Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]
%o A342532 (PARI) \\ here gf gives A106351 as g.f.
%o A342532 gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
%o A342532 seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ _Andrew Howroyd_, Apr 16 2021
%Y A342532 The strictly decreasing version appears to be A064428 (odd-length: A001522).
%Y A342532 The equal version is A065608 (A342527 with odds).
%Y A342532 The weakly decreasing version is A114921 (A342528 with odds).
%Y A342532 Including odds gives A224958.
%Y A342532 A000726 counts partitions with alternating parts unequal.
%Y A342532 A325545 counts compositions with distinct first differences.
%Y A342532 A342529 counts compositions with distinct first quotients.
%Y A342532 Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.
%K A342532 nonn
%O A342532 0,4
%A A342532 _Gus Wiseman_, Mar 28 2021
%E A342532 Terms a(24) and beyond from _Andrew Howroyd_, Apr 16 2021