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A303980 a(n) is the number of cyclic permutations that admit a [1,1,-1]-gridding.

%I A303980 #22 May 18 2018 20:37:19
%S A303980 1,1,1,2,5,15,42,120,338,952,2671,7494,21035,59115,166432,469560,
%T A303980 1327802,3763545,10692500,30447858,86894361,248506757,712109662,
%U A303980 2044402512,5879579540,16937048040,48864612667,141179970820,408444645375,1183143522435,3431241484224,9961919944284
%N A303980 a(n) is the number of cyclic permutations that admit a [1,1,-1]-gridding.
%C A303980 a(n) is the number of cyclic permutations that, when written in their one-line notation, is composed of an increasing segment, followed by another increasing segment, followed by a decreasing segment.
%F A303980 a(n) = A027376(n)/2 - Sum_{i=2..n-1} i*A303979(n,i) when n is odd and n>2.
%F A303980 a(n) = (A027376(n)+A006575(n/2))/2 - Sum_{i=2..n-1} i*A303979(n,i) when n is even and n>2.
%o A303980 (PARI) t051168(n, k) = if (n==0, 1, (1/n) * sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d)));
%o A303980 T303979(n, k) = my(t=sum(j=1, k-1, (-1)^(k+j+1)*t051168(n, j))); if (!(n % 2), t += (-1)^(k+1)*sum(j=1, k-1, if (((n-j) % 4) == 2, t051168(n/2, j/2)))); t;
%o A303980 a027376(n) = if(n<1, n==0, sumdiv(n, d, moebius(n/d)*3^d)/n);
%o A303980 a006575(n) = sumdiv( n, d, if ( bitand(d, 1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n);
%o A303980 a(n) = if (n <= 2, 1, res = a027376(n)/2 - sum(i=2, n-1, i*T303979(n,i)); if (!(n%2), res += a006575(n/2)/2); res); \\ _Michel Marcus_, May 16 2018
%Y A303980 Cf. A027376, A303979, A006575.
%K A303980 nonn
%O A303980 0,4
%A A303980 _Kassie Archer_, May 03 2018
%E A303980 More terms from _Michel Marcus_, May 16 2018