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A239895
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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.
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1
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1, 1, 1, 3, 3, 1, 16, 15, 6, 1, 129, 110, 45, 10, 1, 1438, 1104, 435, 105, 15, 1, 20955, 14455, 5334, 1295, 210, 21, 1, 384226, 238536, 81256, 19089, 3220, 378, 28, 1, 8623101, 4834854, 1509246, 335496, 56259, 7056, 630, 36, 1
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OFFSET
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1,4
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COMMENTS
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LINKS
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Kreweras, G. and Dumont, D., Sur les anagrammes alternés. (French) [On alternating anagrams] Discrete Math. 211 (2000), no. 1-3, 103--110. MR1735352 (2000h:05013).
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FORMULA
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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).
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EXAMPLE
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Triangle begins:
1;
1, 1;
3, 3, 1;
16, 15, 6, 1;
129, 110, 45, 10, 1;
1438, 1104, 435, 105, 15, 1;
20955, 14455, 5334, 1295, 210, 21, 1;
384226, 238536, 81256, 19089, 3220, 378, 28, 1;
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MATHEMATICA
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m = 10(*terms of A218827 for m-1 rows*); matc = Array[0&, {m, m}];
(* The function BellMatrix is defined in A264428.*)
a366[n_] := (-2^(-1))^(n - 2)*Sum[Binomial[n, k]*(1 - 2^(n + k + 1))* BernoulliB[n + k + 1], {k, 0, n}];
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]] ]];
c[n_] := a366[n + 1] - If[n == 1, 0, Sum[ci[n, i], {i, 2, n}]]
T = Rest /@ BellMatrix[c[# + 1]&, m] // Rest;
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PROG
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# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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