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A258498
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Number of words of length 2n such that the index set of occurring letters is {1, 2, ..., k}, all letters are introduced in ascending order, and the words can be built by repeatedly inserting doublets into the initially empty word.
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4
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1, 1, 3, 15, 105, 933, 9988, 124449, 1761287, 27813479, 483482018, 9153385959, 187129080977, 4102129113670, 95861136747795, 2376234441556411, 62216635372018209, 1714347701138957189, 49553280367466054768, 1498300016807379304877, 47270249397381096576643
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ Bell(n-1)*Catalan(n) ~ n^n * exp(n/LambertW(n)-1-n) * 4^n / (sqrt(Pi) * sqrt(1+LambertW(n)) * LambertW(n)^(n-1) * n^(5/2)). - Vaclav Kotesovec, Jun 02 2015
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EXAMPLE
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a(3) = 15: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba, aabbcc, aabccb, abbacc, abbcca, abccba.
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
end:
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..25);
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MATHEMATICA
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A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
T[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];
a[n_] := Sum[T[n, k], {k, 0, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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