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A258498 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. 4

%I

%S 1,1,3,15,105,933,9988,124449,1761287,27813479,483482018,9153385959,

%T 187129080977,4102129113670,95861136747795,2376234441556411,

%U 62216635372018209,1714347701138957189,49553280367466054768,1498300016807379304877,47270249397381096576643

%N 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.

%H Alois P. Heinz, <a href="/A258498/b258498.txt">Table of n, a(n) for n = 0..447</a>

%F a(n) = Sum_{k=0..n} A256117(n,k).

%F 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

%e a(3) = 15: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba, aabbcc, aabccb, abbacc, abbcca, abccba.

%p A:= proc(n, k) option remember; `if`(n=0, 1, k/n*

%p add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))

%p end:

%p T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):

%p a:= n-> add(T(n, k), k=0..n):

%p seq(a(n), n=0..25);

%t 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 T[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];

%t a[n_] := Sum[T[n, k], {k, 0, n}];

%t a /@ Range[0, 25] (* _Jean-Fran├žois Alcover_, Jan 01 2021, after _Alois P. Heinz_ *)

%Y Row sums of A256117.

%Y Cf. A294603, A321031.

%K nonn

%O 0,3

%A _Alois P. Heinz_, May 31 2015

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Last modified April 19 15:00 EDT 2021. Contains 343116 sequences. (Running on oeis4.)