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%I #12 Aug 03 2019 14:20:03
%S 1,1,3,16,129,1438,20955,384226,8623101,231978454,7359117591,
%T 271673905642,11543742745689,559348370431630,30659822500574739,
%U 1887796293833267746,129757032076160998677,9900820197631733600518,834421415151529202479023,77318409826165250051727514
%N Number of indecomposable (by shuffling) alternating n-anagrams.
%C Alternating anagrams enumeration is related to A000366 by a(n) = A000366(n+1).
%C For n>2, a(n) are alternatively congruent to 3 and -2 modulo 18.
%H G. Kreweras and J. Barraud, <a href="http://dx.doi.org/10.1006/eujc.1997.0161">Anagrammes alternés</a>, European Journal of Combinatorics,Volume 18, Issue 8, November 1997, Pages 887-891.
%H G. Kreweras and D. Dumont, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00238-1">Sur les anagrammes alternés</a>, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110.
%t m = 20(*terms*); matc = Array[0&, {m, m}];
%t a366[n_] := (-2^(-1))^(n - 2)*Sum[Binomial[n, k]*(1 - 2^(n + k + 1))* BernoulliB[n + k + 1], {k, 0, n}];
%t ci[n_, k_] := ci[n, k] = Module[{v}, If[matc[[n, k]] == 0, If[n == k, v = 1, If[k == 1, v = c[n], v = Sum[Binomial[n - 1, i - 1]*c[i]*ci[n - i, k - 1], {i, 1, n - k + 1}]]]; matc[[n, k]] = v]; Return[matc[[n, k]]]];
%t a[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]];
%t Array[a, m] (* _Jean-François Alcover_, Aug 03 2019, from PARI *)
%o (PARI) a000366(n)= {return((-1/2)^(n-2)*sum(k=0, n, binomial(n, k)*(1-2^(n+k+1))*bernfrac(n+k+1)));}
%o ci(n,k) = {if (matc[n,k] == 0, if (n==k, v = 1, if (k==1, v = c(n), v = sum(i=1, n-k+1, binomial(n-1,i-1)*c(i)*ci(n-i,k-1)););); matc[n,k] = v;); return(matc[n,k]);}
%o c(n) = {if (n==1, return(a000366(n+1)), return(a000366(n+1) - sum(i=2, n, ci(n,i))));}
%o allc(m) = {matc = matrix(m,m); for (i=1, m, print1(c(i), ", "););}
%Y Cf. A000366, A218826. First column of A239895.
%K nonn
%O 1,3
%A _Michel Marcus_, Nov 07 2012
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