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A239895 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k components.

%I A239895 #30 Mar 27 2020 13:57:20
%S A239895 1,1,1,3,3,1,16,15,6,1,129,110,45,10,1,1438,1104,435,105,15,1,20955,
%T A239895 14455,5334,1295,210,21,1,384226,238536,81256,19089,3220,378,28,1,
%U A239895 8623101,4834854,1509246,335496,56259,7056,630,36,1
%N A239895 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of alternating anagrams on n letters (of length 2n) which are decomposable into at most k components.
%C A239895 The Bell transform of A218827(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 17 2016
%H A239895 Kreweras, G. and Dumont, D., <a href="http://dx.doi.org/10.1016/S0012-365X(99)00238-1">Sur les anagrammes alternés</a>. (French) [On alternating anagrams] Discrete Math. 211 (2000), no. 1-3, 103--110. MR1735352 (2000h:05013).
%F A239895 T(n,k) = C(n-1,0)*c(1)*T(n-1,k-1) + C(n-1,1)*c(2)*T(n-2,k-1) + ... + C(n-1,n-1)*c(n-k+1)*T(k-1,k-1), where c(i) = A218827(i).
%e A239895 Triangle begins:
%e A239895        1;
%e A239895        1,      1;
%e A239895        3,      3,     1;
%e A239895       16,     15,     6,     1;
%e A239895      129,    110,    45,    10,    1;
%e A239895     1438,   1104,   435,   105,   15,   1;
%e A239895    20955,  14455,  5334,  1295,  210,  21,  1;
%e A239895   384226, 238536, 81256, 19089, 3220, 378, 28, 1;
%t A239895 m = 10(*terms of A218827 for m-1 rows*); matc = Array[0&, {m, m}];
%t A239895 (* The function BellMatrix is defined in A264428.*)
%t A239895 a366[n_] := (-2^(-1))^(n - 2)*Sum[Binomial[n, k]*(1 - 2^(n + k + 1))* BernoulliB[n + k + 1], {k, 0, n}];
%t A239895 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 A239895 c[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]]
%t A239895 T = Rest /@ BellMatrix[c[# + 1]&, m] // Rest;
%t A239895 Table[T[[n, k]], {n, 1, m - 1}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 03 2019 *)
%o A239895 (Sage) # uses[bell_matrix from A264428, A218827]
%o A239895 # Adds a column 1,0,0,0,... at the left side of the triangle.
%o A239895 A239895_generator = lambda n: A218827(n+1)
%o A239895 bell_matrix(A239895_generator, 9) # _Peter Luschny_, Jan 17 2016
%Y A239895 Row sums are A000366. First column is A218827.
%K A239895 nonn,tabl,more
%O A239895 1,4
%A A239895 _N. J. A. Sloane_, Apr 04 2014
%E A239895 More terms from _Peter Luschny_, Jan 17 2016