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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.
1

%I #30 Mar 27 2020 13:57:20

%S 1,1,1,3,3,1,16,15,6,1,129,110,45,10,1,1438,1104,435,105,15,1,20955,

%T 14455,5334,1295,210,21,1,384226,238536,81256,19089,3220,378,28,1,

%U 8623101,4834854,1509246,335496,56259,7056,630,36,1

%N 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 The Bell transform of A218827(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 17 2016

%H 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 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 Triangle begins:

%e 1;

%e 1, 1;

%e 3, 3, 1;

%e 16, 15, 6, 1;

%e 129, 110, 45, 10, 1;

%e 1438, 1104, 435, 105, 15, 1;

%e 20955, 14455, 5334, 1295, 210, 21, 1;

%e 384226, 238536, 81256, 19089, 3220, 378, 28, 1;

%t m = 10(*terms of A218827 for m-1 rows*); matc = Array[0&, {m, m}];

%t (* The function BellMatrix is defined in A264428.*)

%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 c[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]]

%t T = Rest /@ BellMatrix[c[# + 1]&, m] // Rest;

%t Table[T[[n, k]], {n, 1, m - 1}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 03 2019 *)

%o (Sage) # uses[bell_matrix from A264428, A218827]

%o # Adds a column 1,0,0,0,... at the left side of the triangle.

%o A239895_generator = lambda n: A218827(n+1)

%o bell_matrix(A239895_generator, 9) # _Peter Luschny_, Jan 17 2016

%Y Row sums are A000366. First column is A218827.

%K nonn,tabl,more

%O 1,4

%A _N. J. A. Sloane_, Apr 04 2014

%E More terms from _Peter Luschny_, Jan 17 2016